Refractive optical elements for generating quasi-non-diffracting laser beams and associated methods

Abstract

A method for processing a transparent workpiece (160) includes generating a plurality of beamlets from a laser beam (12) by transmitting the laser beam through a first refractive optical element (144A). The method also includes generating a plurality of quasi-non- diffracting laser beams from the plurality of beamlets by transmitting the plurality of beamlets through a second refractive optical element (144B). At least 50% of areas of phase distributions individually applied by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4 rad / I1m2. The method also includes directing the plurality of quasi-non-diffracting laser beams into the transparent workpiece simultaneously at different impingement locations on an impingement surface to form a plurality of defects extending at angles relative to each other in a defect plane.

Description

Attorney Docket No.: SP24-029_PCT REFRACTIVE OPTICAL ELEMENTS FOR GENERATING QUASI-NON- DIFFRACTING LASER BEAMS AND ASSOCIATED METHODS CROSS-REFERENCE TO RELATED APPLICATIONS

[0001] This application claims the benefit of priority under 35 U.S.C. § 119 of U.S. Provisional Application Serial No.63 / 553,710, filed on February 15, 2024, the content of which is relied upon and incorporated herein by reference in its entirety. BACKGROUND Field

[0002] The present specification generally relates to refractive optical elements for generatingquasi-non-diffracting laser beams. More particularly, the present specification relates to using the generated quasi-non-diffracting laser beams for laser processing transparent workpieces to generate non-square edges. Technical Background

[0003] The area of laser processing of materials encompasses a wide variety of applications thatinvolve cutting, drilling, milling, welding, melting, etc. of different types of materials. Among these processes, one that is of particular interest is cutting or separating different types of transparent substrates in a process that may be utilized in the production of materials such as glass, sapphire, or fused silica for thin film transistors (TFT) or display materials for electronic devices.

[0004] From process development and cost perspectives there are many opportunities forimprovement in cutting and separating glass substrates. It is of great interest to have a faster, cleaner, cheaper, more repeatable, and more reliable method of separating glass substrates than what is currently practiced in the market. Many methods of separating glass substrates result in edges with square corners that are prone to breakage and are often processed to have bevels or to be rounded to minimize the chance of breakage. Currently, the non-square edges are often accomplished using mechanical means, such as mechanical grinding and polishing. However, the processes generate glass dust and particles, which must be cleaned by additional process steps involving washing or chemical treatments. Accordingly, a need exists for alternative improvedAttorney Docket No.: SP24-029_PCT methods for separating glass substrates which replace the conventional edge finishing process with a particle free and high throughput process. SUMMARY

[0005] An aspect (1) of the present disclosure pertains to an optical system includes a firstrefractive optical element configured to separate a laser beam into a plurality of beamlets each propagating in a different propagation direction, a second refractive optical element comprising a plurality of beamlet regions, each beamlet region being configured to apply a different phase distribution to a separate one of the plurality of beamlets to convert the plurality of beamlets into a plurality of quasi-non-diffracting laser beams, wherein: each of the plurality of quasi-non- diffracting laser beams propagates along a separate propagation axis, and at least 50% of areas of phase distributions applied individually by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2.

[0006] An aspect (2) of the present disclosure pertains to an optical system according to theaspect (1), wherein the first refractive optical element comprises a plurality of concentric regions configured to separate different spatial portions of the laser beam into the plurality of beamlets.

[0007] An aspect (3) of the present disclosure pertains to an optical system according to theaspect (2), wherein the plurality of concentric regions comprises: a central region configured to generate a first beamlet from a central portion of the laser beam, a first annular region configured to generate a second beamlet from a first annular portion of the laser beam circumferentially surrounding the central portion, and a second annular region configured to generate a third beamlet from a second annular portion of the laser beam circumferentially surrounding the first annular portion.

[0008] An aspect (4) of the present disclosure pertains to an optical system according to any ofthe aspects (2)-(3), wherein each of the plurality of concentric regions applies a different linear phase to each of the different spatial portions so that each of the plurality of beamlets is incident on one of the beamlet regions of the second refractive optical element.Attorney Docket No.: SP24-029_PCT

[0009] An aspect (5) of the present disclosure pertains to an optical system according to theaspect (4), wherein centers of each of the plurality of beamlet regions form a line in the transverse optical plane.

[0010] An aspect (6) of the present disclosure pertains to an optical system according to theaspect (5), wherein the plurality of beamlet regions comprises two or more beamlet regions having non-equal surface areas.

[0011] An aspect (7) of the present disclosure pertains to an optical system according to theaspect (6), wherein the two or more beamlet regions comprises: a first beamlet region having a smallest projected surface area of the plurality of beamlet regions and disposed at a first end of the plurality of beamlet regions, a third beamlet region having a largest projected surface area of the plurality of beamlet regions and disposed at a second end of the plurality of beamlet regions, and a second beamlet region being disposed between the first beamlet region and the third beamlet region and having a projected surface area between that of the first beamlet region and the third beamlet region, wherein the plurality of concentric regions are configured such that the central portion is directed to the first beamlet region, the first annular portion is directed to the second beamlet region, and the second annular portion is directed to the third beamlet region.

[0012] An aspect (8) of the present disclosure pertains to an optical system according to theaspect (7), wherein the second beamlet region is oriented within 1° of normal relative to the optical axis, and the first beamlet region and the third beamlet region are inclined with respect to the second beamlet region at inclination angles of at least 0.25°.

[0013] An aspect (9) of the present disclosure pertains to an optical system according to any ofthe aspects (1)-(8), wherein the first refractive optical element is constructed such that each of the plurality of beamlets comprises substantially equal power.

[0014] An aspect (10) of the present disclosure pertains to an optical system according to any ofthe aspects (1)-(9), wherein each of the plurality of beamlet regions is configured to apply a corrective phase configured to counteract the phase imparted by the first refractive optical element and a quasi-non-diffractive phase to each beamlet.Attorney Docket No.: SP24-029_PCT

[0015] An aspect (11) of the present disclosure pertains to an optical system according to any ofthe aspects (1)-(10), wherein at least one of the plurality of beamlet regions are configured to apply a linear phase to at least one of the plurality of beamlets so that at least two of the plurality of quasi-non-diffracting beams propagate in different propagation directions downstream of the second refractive optical element.

[0016] An aspect (12) of the present disclosure pertains to an optical system according to any ofthe aspects (1)-(11), further comprising a lens assembly configured to relay the plurality of quasi- non-diffracting beams downstream from the first refractive optical element and the second refractive optical element.

[0017] An aspect (13) of the present disclosure pertains to an optical system according to theaspect (12), further comprising a transparent workpiece disposed downstream from the lens assembly, wherein the plurality of focal lines are formed in the transparent workpiece and generate defects extending through the thickness of the transparent workpiece at angles relative to one another.

[0018] An aspect (14) of the present disclosure pertains to an optical system according to any ofthe aspects (1)-(13), wherein the first refractive optical element and the second refractive optical element are disposed in a common housing configured to rotate about an optical axis of the optical system to rotate the orientation of the plurality of quasi-non-diffracting laser beams.

[0019] An aspect (15) of the present disclosure pertains to an optical system for forming defectsin a transparent workpiece, the optical system including: a laser beam source configured to emit a laser beam; a first refractive optical element configured to separate the laser beam into a plurality of beamlets, a second refractive optical element comprising a plurality of beamlet regions, each beamlet region being positioned to receive one of the plurality of beamlets and apply a phase distribution thereto to convert that beamlet into a quasi-non-diffracting laser beam, wherein at least 50% of areas of phase distributions applied individually by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2; and a lens system configured to relay each quasi-non-diffracting laser beam into the transparent workpiece to generate defects withAttorney Docket No.: SP24-029_PCT defect segments that extend through the transparent workpiece at angles relative to one another in a defect plane.

[0020] An aspect (16) of the present disclosure pertains to an optical system according to theaspect (15), wherein the first refractive optical element comprises a plurality of concentric regions configured to separate different spatial portions of the laser beam into the plurality of beamlets.

[0021] An aspect (17) of the present disclosure pertains to an optical system according to theaspect (16), wherein the plurality of concentric regions comprises: a central region configured to generate a first beamlet from a central portion of the laser beam, a first annular region configured to generate a second beamlet from a first annular portion of the laser beam circumferentially surrounding the central portion, and a second annular region configured to generate a third beamlet from a second annular portion of the laser beam circumferentially surrounding the first annular portion.

[0022] An aspect (18) of the present disclosure pertains to an optical system according to any ofthe aspects (16)-(17), wherein each of the plurality of concentric regions applies a different linear phase to each of the different spatial portions so that each of the plurality of beamlets is incident on one of the beamlet regions of the second refractive optical elements.

[0023] An aspect (19) of the present disclosure pertains to an optical system according to theaspect (18), wherein centers of each of the plurality of beamlet regions form a line in the transverse optical plane.

[0024] An aspect (20) of the present disclosure pertains to an optical system according to theaspect (19), wherein the plurality of beamlet regions comprises: a first beamlet region having a smallest projected surface area of the plurality of beamlet regions and disposed at a first end of the plurality of beamlet regions, a third beamlet region having a largest projected surface area of the plurality of beamlet regions and disposed at a second end of the plurality of beamlet regions, and a second beamlet region being disposed between the first beamlet region and the third beamlet region and having a projected surface area between that of the first beamlet region and the third beamlet region.Attorney Docket No.: SP24-029_PCT

[0025] An aspect (21) of the present disclosure pertains to an optical system according to theaspect (20), wherein the plurality of concentric regions are configured such that the central portion is directed to the first beamlet region, the first annular portion is directed to the second beamlet region, and the second annular portion is directed to the third beamlet region.

[0026] An aspect (22) of the present disclosure pertains to an optical system according to theaspect (21), wherein: the second beamlet region is oriented within 1° of normal relative to the optical axis, and the first beamlet region and the third beamlet region are inclined with respect to the third beamlet region at inclination angles of at least 0.25°.

[0027] An aspect (23) of the present disclosure pertains to an optical system according to theaspect (22), wherein the first refractive optical element is constructed such that each of the plurality of beamlets comprises substantially equal power.

[0028] An aspect (24) of the present disclosure pertains to an optical system according to any ofthe aspects (15)-(23), wherein each of the plurality of beamlet regions is configured to apply a corrective phase configured to counteract the phase imparted by the first refractive optical element and a quasi-non-diffractive phase to each beamlet.

[0029] An aspect (25) of the present disclosure pertains to an optical system according to any ofthe aspects (15)-(24), wherein at least one of the plurality of beamlet regions are configured to apply a linear phase to at least one of the plurality of beamlets so that at least two of the plurality of quasi-non-diffracting beams propagate in different propagation directions downstream of the second refractive optical element.

[0030] An aspect (26) of the present disclosure pertains to an optical system according to any ofthe aspects (15)-(25), wherein: the first refractive optical element, the second optical element, and the lens assembly are configured to generate a first quasi-non-diffracting laser beam, a second quasi-non-diffracting laser beam, and a third quasi-non-diffracting laser beam in the transparent workpiece, the first quasi-non-diffracting beam generates a first defect segment in the defect plane that extends from an impingement surface into the transparent workpiece at a first chamfer angle relative to a surface normal of the impingement surface, the second quasi-non-diffracting laser beam generates a second defect segment in the defect plane that extends from a termination pointAttorney Docket No.: SP24-029_PCT of the first defect into the transparent workpiece at a second chamfer angle relative to the surface normal, and the third quasi-non-diffracting laser beam generates a third defect segment in the defect plane that extends from a termination point of the second defect at a third chamfer angle relative to the surface normal.

[0031] An aspect (27) of the present disclosure pertains to an optical system according to theaspect (26), wherein the first chamfer angle equals the third chamfer angle and the second chamfer angle is less than 1°.

[0032] An aspect (28) of the present disclosure pertains to an optical system according to theaspect (27), wherein the first chamfer angle and the third chamfer angle are greater than 5°.

[0033] An aspect (29) of the present disclosure pertains to an optical system according to any ofthe aspects (15)-(28), wherein the first refractive optical element and the second refractive optical element are disposed in a common housing configured to rotate about an optical axis of the optical system to rotate the defect plane within the transparent workpiece.

[0034] An aspect (30) of the present disclosure pertains to a method for processing a transparentworkpiece, the method including: generating a plurality of beamlets from a laser beam by transmitting the laser beam through a first refractive optical element; generating a plurality of quasi-non-diffracting laser beams from the plurality of beamlets by transmitting the plurality of beamlets through a second refractive optical element, wherein at least 50% of areas of phase distributions individually applied by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2; directing the plurality of quasi-non-diffracting laser beams into the transparent workpiece simultaneously at different impingement locations on an impingement surface to form a plurality of defects, with each defect comprising a plurality of defect segments extending at angles relative to each other in a defect plane.

[0035] An aspect (31) of the present disclosure pertains to a method according to the aspect (30),further comprising translating the plurality of quasi-diffracting laser beams and the transparent workpiece relative to each other to move the defect plane and form a plurality of defects along a contour.Attorney Docket No.: SP24-029_PCT

[0036] An aspect (32) of the present disclosure pertains to a method according to the aspect (31),further comprising rotating the first refractive optical element and the second refractive optical element in conjunction with one another to rotate a linear arrangement of the plurality of quasi- non-diffracting laser beams and provide the contour with a curved shape.

[0037] An aspect (33) of the present disclosure pertains to an optical system according to any ofthe aspects (30)-(32), further comprising separating the transparent workpiece along the contour to form a separated glass article.

[0038] An aspect (34) of the present disclosure pertains to a method according to any of theaspects (30)-(33), wherein the plurality of quasi-non-diffracting laser beams forms a first laser beam focal line, a second laser beam focal line, and a third laser beam focal line in the transparent workpiece to generate the plurality of defects, wherein: the first laser beam focal line generates a first defect in the defect plane that extends from an impingement surface into the transparent workpiece at a first chamfer angle relative to a surface normal of the impingement surface, the second laser beam focal line generates a second defect in the defect plane that extends from a termination point of the first defect into the transparent workpiece at a second chamfer angle relative to the surface normal, and the third laser beam focal line generates a third defect in the defect plane that extends from a termination point of the second defect at a third chamfer angle relative to the surface normal.

[0039] An aspect (35) of the present disclosure pertains to a method according to the aspect (34),wherein the first chamfer angle equals the third chamfer angle and the second chamfer angle is less than 1°.

[0040] An aspect (36) of the present disclosure pertains to a method according to the aspect (35),wherein the first chamfer angle and the third chamfer angle are greater than 5°.

[0041] Additional features and advantages of the processes and systems described herein will beset forth in the detailed description which follows, and in part will be readily apparent to those skilled in the art from that description or recognized by practicing the embodiments described herein, including the detailed description which follows, the claims, as well as the appended drawings.Attorney Docket No.: SP24-029_PCT

[0042] It is to be understood that both the foregoing general description and the followingdetailed description describe various embodiments and are intended to provide an overview or framework for understanding the nature and character of the claimed subject matter. The accompanying drawings are included to provide a further understanding of the various embodiments, and are incorporated into and constitute a part of this specification. The drawings illustrate the various embodiments described herein, and together with the description serve to explain the principles and operations of the claimed subject matter. BRIEF DESCRIPTION OF THE DRAWINGS

[0043] The embodiments set forth in the drawings are illustrative and exemplary in nature andnot intended to limit the subject matter defined by the claims. The following detailed description of the illustrative embodiments can be understood when read in conjunction with the following drawings, where like structure is indicated with like reference numerals and in which:

[0044] FIG. 1A schematically depicts a perspective view of an embodiment of laser forming acontour of defects in a transparent workpiece, according to one or more embodiments described herein;

[0045] FIG. 1B schematically depicts a side view of an embodiment of laser forming the contourof defects in the transparent workpiece, according to one or more embodiments described herein;

[0046] FIG. 2A schematically depicts an example laser beam downstream an axicon, accordingto one or more embodiments described herein;

[0047] FIG. 2B schematically depicts another example laser beam downstream an axicon,according to one or more embodiments described herein;

[0048] FIG. 3 schematically depicts an optical assembly for laser processing a transparentworkpiece including a beam source, a first refractive optical element, a second refractive optical element, and a lens assembly, according to one or more embodiments described herein;Attorney Docket No.: SP24-029_PCT

[0049] FIG. 4A depicts a first phase mask that may be implemented by the first and secondrefractive optical elements depicted in FIG.4Bto generate a plurality of beamlets from an initial laser beam, according to one or more embodiments of the present disclosure;

[0050] FIG. 4B depicts a second phase mask that may be implemented by the first and secondrefractive optical elements depicted in FIG.4Bto generate a plurality of beams that are quasi-non- diffracting from the plurality of beamlets generated via the first phase mask depicted in FIG.4A, according to one or more embodiments of the present disclosure;

[0051] FIG. 4C depicts another first phase mask that may be implemented by the first and secondrefractive optical elements depicted in FIG.4B to generate a plurality of beamlets from an initial laser beam, according to one or more embodiments of the present disclosure;

[0052] FIG. 4D depicts another second phase mask that may be implemented by the first andsecond refractive optical elements depicted in FIG. 4B to generate a plurality of beams that are quasi-non-diffracting from the plurality of beamlets generated via the first phase mask depicted in FIG.4C, according to one or more embodiments of the present disclosure;

[0053] FIG. 5A schematically depicts a plurality of concentric regions of a first phase mask thatmay generate beamlets that each propagate along an independent propagation axis from different spatial portions of an initial laser beam, according to one or more embodiments of the present disclosure;

[0054] FIG. 5B schematically depicts spatial portions of an initial laser beam both prior to beingincident on the first phase mask represented in FIG. 5A and downstream the first phase mask, according to one or more embodiments of the present disclosure;

[0055] FIG. 5C depicts an initial arrangement of the spatial portions of the initial laser beamshown in FIG. 5A and a desired arrangement of beamlets at a second phase mask, according to one or more embodiments of the present disclosure;

[0056] FIG. 6A depicts a first phase mask computed by the methods described herein that maybe implemented by the first refractive optical element of the optical assembly depicted in FIG. 3Attorney Docket No.: SP24-029_PCT to generate a plurality of beamlets from an initial laser beam, according to one or more embodiments of the present disclosure;

[0057] FIG. 6B depicts a second phase mask computed by the methods described herein thatmay be implemented by the second refractive optical element of the optical assembly depicted in FIG.3 to generate a plurality of beams that are quasi-non-diffracting from the plurality of beamlets produced via the first phase mask depicted in FIG.6A, according to one or more embodiments of the present disclosure;

[0058] FIG. 6C depicts another first phase mask computed by the methods described herein thatmay be implemented the first refractive optical element of the optical assembly depicted in FIG.3 to generate a plurality of beamlets from an initial laser beam, according to one or more embodiments of the present disclosure;

[0059] FIG. 6D depicts another second phase mask computed by the methods described hereinthat may be implemented by the second refractive optical element of the optical assembly depicted in FIG. 3 to generate a plurality of beams that are quasi-non-diffracting from the plurality of beamlets when used in conjunction with the first phase mask depicted in FIG. 6C, according to one or more embodiments of the present disclosure;

[0060] FIG. 7 is a surface plot of unwrapped versions of the first phase masks depicted in FIGS.4A, 4C, 6A, and 6C, according to one or more embodiments of the present disclosure;

[0061] FIGS. 8A – 8D are histograms of change in slope of phase for pixels of the first phasemasks depicted in FIGS.4A, 4C, 6A, and 6C, according to one or more embodiments of the present disclosure;

[0062] FIG. 8E is a plot of cumulative pixel counts as a function of change in phase slope forthe first phase masks depicted in FIGS. 4A, 4C, 6A, and 6C, according to one or more embodiments of the present disclosure;

[0063] FIG. 8F is a plot of cumulative pixel counts as a function of change in phase slope forthe second phase masks depicted in FIGS. 4B, 4D, 6B, and 6C, according to one or more embodiments of the present disclosure;Attorney Docket No.: SP24-029_PCT

[0064] FIG. 9A is a plot of simulated average relative intensity of a laser beam focal linegenerated by the pairs of phase masks depicted in FIGS. 4A-4D and 6A-6D as a function of diagonal mask offset (in terms of pixels), according to one or more embodiments of the present disclosure;

[0065] FIG. 9B is a plot of simulated average relative intensity of a laser beam focal linegenerated by the pairs of phases masks depicted in FIGS. 4A-4D and 6A-6D as a function of convolution kernel width applied to the phase masks to simulate feature blurring, according to one or more embodiments of the present disclosure;

[0066] FIG. 10A is a surface height map for a first refractive optical element, according to oneor more embodiments of the present disclosure;

[0067] FIG. 10B is a surface height map for a second refractive optical element, according toone or more embodiments of the present disclosure;

[0068] FIG. 11 depicts the surface height map of FIG. 10A with the background removed andoverlaid on a polar point grid to facilitate fabrication of the second refractive optical element, according to one or more embodiments of the present disclosure;

[0069] FIG. 12 is surface height profile of a variable surface of an example first refractive opticalelement fabricated via the methods described herein, according to one or more embodiments of the present disclosure;

[0070] FIG. 13 is a surface height profile of a variable surface of an example second refractiveoptical element fabricated via the methods described herein, according to one or more embodiments of the present disclosure;

[0071] FIG. 14 schematically depicts the lateral arrangement of the first beam, the second beam,and the third beam of the laser beam combination at a surface of a second refractive optical element, according to one or more embodiments shown and described herein;

[0072] FIG. 15A graphically depicts the relative intensity of laser pulses within an exemplarypulse burst vs. time, according to one or more embodiments described herein;Attorney Docket No.: SP24-029_PCT

[0073] FIG. 15B graphically depicts relative intensity of laser pulses vs. time within anotherexemplary pulse burst, according to one or more embodiments described herein;

[0074] FIG. 15C graphically depicts intensity of laser pulses vs. distance, according to one ormore embodiments described herein;

[0075] FIG. 16A schematically depicts a side view of a transparent workpiece having a contourof C-chamfered defects, according to one or more embodiments described herein; and

[0076] FIG. 16B schematically depicts a side view of two separated articles formed from thetransparent workpiece of FIG. 16A, each separated article comprising a C-chamfered edge, according to one or more embodiments shown and described herein. DETAILED DESCRIPTION

[0077] Reference will now be made in detail to embodiments of processes for laser processingtransparent workpieces, such as glass workpieces, examples of which are illustrated in the accompanying drawings. Whenever possible, the same reference numerals will be used throughout the drawings to refer to the same or like parts.

[0078] According to one or more embodiments described herein, a transparent workpiece maybe laser processed to form a contour in the transparent workpiece that comprises a series of defects along a line of intended separation for separating the transparent workpiece into two or more separated articles. In embodiments, each of the defects comprise a first defect segment and a second defect segment, where the first defect segment comprises a chamfer angle of greater than 5° such that, after separation of the transparent workpiece along the contour, the resultant separated articles comprise an angled edge portion having an edge angle of greater than 5°. In the embodiments described herein, the chamfer angle of the defect segments is measured relative to a plane orthogonal to an impingement surface of the transparent workpiece in either a clockwise or counterclockwise rotational direction. Moreover, in some embodiments, the second defect segment comprises a chamfer angle having a magnitude of less than 1°, such as 0° and the defects also include a third defect segment which comprises a chamfer angle having a magnitude of greater than 5°. Thus, the resultant separated articles may comprise a straight edge portion between twoAttorney Docket No.: SP24-029_PCT opposite angled portions, that is, a C-chamfered edge. Defects may be formed in a transparent workpiece using a low diffracting beam, such as a quasi-non-diffracting beam.

[0079] Certain existing techniques for forming such defect segments, such as those described inInternational Patent Application No. WO 2023 / 009331 A1, entitled “Phase Modified Quasi-Non- Diffracting Laser Beams for Simultaneous High Angle Laser Processing of Transparent Workpieces,” hereby incorporated by reference in its entirety, use phase-altered angled laser beam focal lines. In such existing techniques, a plurality of phase masks are applied to an initial laser beam to form a laser beam combination and form a quasi-non-diffracting laser beam from each laser beam in the laser beam combination. WO 2023 / 009331 A1 describes using either an adaptive phase altering optical element (e.g., a spatial light modulator, an adaptive phase plate, or a deformable mirror) or diffractive optical elements to implement the phase masks needed to transform the initial laser beam into a plurality of quasi-non-diffracting beams at the workpiece. These components suffer from various drawbacks that may hinder adoption in many industrial applications. Programmable optical elements, such as spatial light modulators, adaptive phase plates, and digital micromirrors, suffer from relatively low power efficiency and are sensitive to environmental variables, leading to process instability. Diffractive optical elements rely on quantized thickness variations on the order of the wavelength of light being used in the laser processing, which significantly limits the design space for the optics and the phase masks that can be employed and leads to alignment sensitivities.

[0080] The systems and methods of the present disclosure address these deficiencies associatedwith using programmable optical elements or diffractive optical elements to implement phase masks for forming a plurality of quasi-non-diffracting beams from an initial laser beam by replacing such elements with static refractive optical elements. That is, static, non-programmable refractive optical elements are used to implement the necessary wavefront perturbations on the light from the initial laser beam to form a plurality of defect segments extending at angles relative to one another in the transparent workpiece. To use such a static refractive optical element architecture to perform the phase transformations described herein, certain limitations are imposed on the phase distributions imparted by the refractive optical elements. The phase distributions imparted by the diffractive optical elements or adaptive phase altering optical elements described in WO 2023 / 009331 A1 were holographic-type designs relying on sharp phase transitions (largeAttorney Docket No.: SP24-029_PCT phase gradients). Such holographic designs are not easily practicable in refractive form, as existing grinding and polishing methods used to form refractive optical elements are typically incapable of providing very sharp phase transitions.

[0081] The refractive optical elements described herein apply quasi-continuous phasedistributions to the laser beam, with these quasi-continuous phase distributions being implemented by thickness variations therein that are relatively smooth (i.e., possess relatively small spatial derivatives as compared to the previously described holographic-type designs). As used herein, the term “quasi-continuous phase distribution” describes a phase distribution imparted by an individual refractive optical element and means that at least 50% of the area of the phase distribution applied individually by a refractive optical element to light from a laser beam is associated with a change in phase slope that is less than 1.7x10-4rad / µm2. The previously described holographic-type designs do not have such a large portion of the phase distribution associated with such a low value for in change in phase slope. Such smooth thickness profiles enable the manufacturing of the refractive optics described herein using commercially available fabrication equipment.

[0082] To provide such phase distributions with such smoothly varying phases, the presentdisclosure uses a particular approach for designing an optical system including a first refractive optical element and a second refractive optical element. Operationally, the first refractive optical element is configured to separate a laser beam from a beam source into a plurality of beamlets, with each beamlet propagating in a different propagation direction; and the second refractive optical element comprises a plurality of beamlet regions, with each beamlet region being configured to apply a different phase distribution to a separate one of the plurality of beamlets to convert the plurality of beamlets into a plurality of quasi-non-diffracting laser beams. To apply the needed constraints most effectively to the phase computation algorithms described herein to obtain quasi-continuous phase distributions for each of the first and second refractive optical elements, it has been found that an approach can be used where first refractive optical element comprises a plurality of radially symmetric concentric regions configured to separate different spatial portions of the laser beam into the plurality of beamlets. Each concentric regions forms a separate beamlet from a different spatial portion of the incoming laser beam, with each beamlet propagating in a separate propagation axis to be incident on a different beamlet region of theAttorney Docket No.: SP24-029_PCT second optical element. This construction has been found to aid in model convergence in the computation of the phase distribution for the second refractive optical element that meets the minimal slope variance requirement described herein. More details on the restraints applied to the phase computation algorithms are provided herein.

[0083] The implementation of the first refractive optical element using the plurality of radiallysymmetric concentric regions described herein has been found to aid in the fabrication thereof using a diamond turning lathe. Particularly, each concentric region is able to be treated separately in the computer-aided manufacturing package to provide different tool paths for each region. It was found that each region was within acceptable operating parameters of the lathe (e.g., avoided unacceptable acceleration spikes). Moreover, the second refractive optical element can be fabricated by reconfiguring the point cloud used in generating the tool path, as described in greater detail herein.

[0084] The refractive optical elements described herein are able to successfully generate aplurality of quasi-non-diffracting laser beams from an initial laser beam and control the propagation direction of each quasi-non-diffracting beam to provide a desired arrangement of laser beam focal lines within a transparent workpiece to provide defect segments along a desired contour with desired orientations relative to one another. It is believed that the refractive optical elements with the quasi-continuous phase profiles described herein provide several advantages over the existing holographic-type designs used in the existing approach. First, refractive optics are believed to have more than 95% optical efficiency (when appropriate anti-reflective coatings are included), while spatial light modulators are less than 80% efficient. Second, the refractive optical elements described herein have continuous, non-quantized slopes, and thus possess a significantly larger design space than diffractive optical elements (more flexibility is provided). Third, compared to spatial light modulated-based beam shaping, non-programmable refractive optics are less sensitive to environmental conditions and are more reliable. Fourth, the quasi-continuous nature of the phase distributions described herein renders the overall system less sensitive to optical misalignment, rendering the overall system more reliable. Fifth, the refractive optics described herein can easily be rotated in conjunction with one another to rotate the arrangement of quasi-non-diffracting beams in conjunction with one another to provide contours with curved corners. In short, the process using the refractive optical elements described herein is moreAttorney Docket No.: SP24-029_PCT efficient, more reliable, and provides more flexibility than previous approaches utilizing holographic-type phase distributions with sharp phase changes.

[0085] As used herein, “laser processing” comprises directing a laser beam onto and / or into atransparent workpiece. In some embodiments, laser processing further comprises translating the laser beam relative to the transparent workpiece or translating the transparent workpiece relative to the laser beam, for example, along a contour line or other pathway. Examples of laser processing include using a laser beam to form a contour comprising a series of defects that extend into the transparent workpiece and / or using an infrared laser beam to heat the transparent workpiece. Laser processing may separate the transparent workpiece along one or more desired lines of separation. However, in some embodiments, additional non-laser steps, such as applying mechanical force, may be utilized to separate the transparent workpiece along one or more desired lines of separation.

[0086] As used herein, the “angular spectrum” of a laser beam refers to the distribution of theFourier spectrum of the laser beam in the spatial frequency domain. In particular, the angular spectrum represents a group of plane waves whose summation recreates the original beam. The angular spectrum may also be referred to as the spatial-frequency distribution of the laser beam. As used herein, a “circular angular spectrum” is an angular spectrum whose peak intensity region forms a circular shape with a radius varying less than 5% relative to a central point of the circular angular spectrum, which is positioned at the propagation direction of the beam.

[0087] As used herein, “beam spot” refers to a cross section of a laser beam (e.g., a beam crosssection) at the impingement location of the laser beam at an impingement surface of a transparent workpiece, i.e., the surface of a transparent workpiece upon which the laser beam is first incident. The beam spot is the cross-section at the impingement location. In the embodiments described herein, the beam spot is sometimes referred to as being “axisymmetric” or “non-axisymmetric.” As used herein, axisymmetric refers to a shape that is symmetric, or appears the same, for any arbitrary rotation angle made about a central axis, and “non-axisymmetric” refers to a shape that is not symmetric for any arbitrary rotation angle made about a central axis. The rotation axis (e.g., the central axis) is most often taken as being the optical axis (axis of propagation) of the laser beam, which is the axis extending in the beam propagation direction, which is referred to herein as the z-direction.Attorney Docket No.: SP24-029_PCT

[0088] As used herein, “upstream” and “downstream” refer to the relative position of two locationsor components along a beam pathway with respect to a beam source. For example, a first component is upstream from a second component if the first component is closer to the beam source along the path traversed by the laser beam than the second component and a first position (location) is upstream from a second position (location) if the first position (location) is closer to the beam source along the path traversed by the laser beam than the second position (location).

[0089] As used herein, “beam pathway” refers to an alterable direction of travel of the laser beam.The beam pathway is determined by the direction of travel of the laser beam, and alters based on the positioning and direction of the laser beam.

[0090] As used herein, “laser beam focal line,” refers to a pattern of interacting (e.g., crossing)light rays of a laser beam that forms a focal region elongated in the beam propagation direction. In conventional laser processing, a laser beam is tightly focused to a focal point. The focal point is the point of maximum intensity of the laser beam and is situated at a focal plane in a transparent workpiece. In the elongated focal region of a focal line, in contrast, the region of maximum intensity of the laser beam extends beyond a point to a line aligned with the beam propagation direction. A focal line is formed by converging light rays that intersect (e.g., cross) to form a continuous series of focal points aligned with the beam propagation direction. The laser beam focal lines described herein are formed using a quasi-non-diffracting beam, mathematically defined in detail below.

[0091] As used herein, “contour line,” corresponds to the set of intersection points of the laserbeam with the incident (impingement) surface of the transparent workpiece resulting from relative motion of the laser beam and the transparent workpiece. A contour line can be linear, angled, polygonal or curved in shape. A contour line can be closed (i.e. defining an enclosed region on the surface of the transparent workpiece) or open (i.e. not defining an enclosed region on the surface of the transparent workpiece). The contour line represents a boundary along which separation of the transparent workpiece into two or more parts is facilitated. Separation occurs spontaneously or with the assistance of external thermal or mechanical energy.

[0092] As used herein, “contour,” refers to a set of defects in a transparent workpiece formed bya laser beam through relative motion of a laser beam and the transparent workpiece along a contourAttorney Docket No.: SP24-029_PCT line. The defects are spaced apart along the contour line and are wholly contained within the interior of the transparent workpiece and / or extend through one or more surfaces into the interior of the transparent workpiece. Defects may also extend through the entire thickness of the transparent workpiece. Separation of the transparent workpiece occurs by connecting defects along the contour, such as, for example, through propagation of a crack.

[0093] As used herein, a “defect” refers to a region of a transparent workpiece that has beenmodified by a laser beam focal line. Defects include regions of a transparent workpiece having a modified refractive index relative to surrounding unmodified regions of the transparent workpiece. Common defects include structurally modified regions such as void spaces, cracks, scratches, flaws, holes, perforations, densifications, or other deformities in the transparent workpiece produced by a laser beam focal line. Defects may also be referred to, in various embodiments herein, as defect lines or damage tracks. A defect or damage track is formed through interaction of a laser beam focal line with the transparent workpiece. As described more fully below, the laser beam focal line is produced by a pulsed laser. A defect at a particular location along the contour line is formed from a focal line produced by a single laser pulse at the particular location, by a pulse burst of sub-pulses at the particular location, or by multiple laser pulses at the particular location. Relative motion of the laser beam and transparent workpiece along the contour line results in multiple defects that form a contour.

[0094] The phrase “transparent workpiece,” as used herein, means a workpiece formed from glass,glass-ceramic or other material which is transparent, where the term “transparent,” as used herein, means that the workpiece has a linear optical absorption of less than 20% per mm of material depth for the specified pulsed laser wavelength. In embodiments, the transparent workpiece has a linear optical absorption less than 10% per mm of material depth for the specified pulsed laser wavelength, or such as less than 1% per mm of material depth for the specified pulsed laser wavelength. Unless otherwise specified, the transparent workpiece has a linear optical absorption of less than about 20% per mm of material depth. The transparent workpiece may have a depth (e.g., thickness) of from about 50 microns (µm) to about 10 mm (such as from about 100 µm to about 5 mm, or from about 0.5 mm to about 3 mm). Transparent workpieces may comprise glass workpieces formed from glass compositions, such as borosilicate glass, soda-lime glass, aluminosilicate glass, alkali aluminosilicate, alkaline earth aluminosilicate glass, alkaline earthAttorney Docket No.: SP24-029_PCT boro-aluminosilicate glass, fused silica, or crystalline materials such as sapphire, silicon, gallium arsenide, or combinations thereof. In some embodiments the transparent workpiece may be strengthened via thermal tempering before or after laser processing the transparent workpiece. In some embodiments, the glass may be ion-exchangeable or ion exchanged, such that the glass composition can undergo ion-exchange or has undergone ion-exchange for glass strengthening before or after laser processing the transparent workpiece. For example, the transparent workpiece may comprise ion exchanged or ion exchangeable glass, such as Corning Gorilla® Glass available from Corning Incorporated of Corning, NY (e.g., code 2318, code 2319, and code 2320). Further, these ion exchangeable or ion exchanged glasses may have coefficients of thermal expansion (CTE) of from about 6 ppm / ºC to about 10 ppm / ºC. Other examples of transparent workpieces may comprise EAGLE XG®and CORNING LOTUSTMavailable from Corning Incorporated of Corning, NY. Moreover, the transparent workpiece may comprise other components which are transparent to the wavelength of the laser, for example, glass ceramics or crystals such as sapphire or zinc selenide.

[0095] In an ion exchange process, ions in a surface layer of the transparent workpiece arereplaced by larger ions having the same valence or oxidation state, for example, by partially or fully submerging the transparent workpiece in an ion exchange bath. Replacing smaller ions with larger ions causes a layer of compressive stress to extend from one or more surfaces of the transparent workpiece to a certain depth within the transparent workpiece, referred to as the depth of layer. The compressive stresses are balanced by a layer of tensile stresses (referred to as central tension) such that the net stress in the glass sheet is zero. The formation of compressive stresses at the surface of the glass sheet makes the glass strong and resistant to mechanical damage and, as such, mitigates catastrophic failure of the glass sheet for flaws, which do not extend through the depth of layer. In some embodiments, smaller sodium ions in the surface layer of the transparent workpiece are exchanged with larger potassium ions. In some embodiments, the ions in the surface layer and the larger ions are monovalent alkali metal cations, such as Li+(when present in the glass), Na+, K+, Rb+, and Cs+. Alternatively, monovalent cations in the surface layer may be replaced with monovalent cations other than alkali metal cations, such as Ag+, Tl+, Cu+, or the like.

[0096] As used herein, the term “quasi-non-diffracting beam” is used to describe a laser beamhaving low beam divergence as mathematically described below. In particular, the laser beam isAttorney Docket No.: SP24-029_PCT used to form a contour of defects in the embodiments described herein. The laser beam has an intensity distribution I(X,Y,Z), where Z is the beam propagation direction of the laser beam, and X and Y are directions orthogonal to the beam propagation direction, as depicted in the figures. The X-direction and Y-direction may also be referred to as cross-sectional directions and the X-Y plane may be referred to as a cross-sectional plane. The coordinates and directions X, Y, and Z are also referred to herein as x, y, and z; respectively. The intensity distribution of the laser beam in a cross-sectional plane may be referred to as a cross-sectional intensity distribution.

[0097] The quasi-non-diffracting laser beam may be formed by transmitting a diffracting laserbeam (such as a Gaussian beam) through the first and second refractive optical elements described herein to modify the phase of the beam, to reduce beam divergence, and to increase Rayleigh range, as mathematically defined below. Example quasi-non-diffracting beams include Gauss- Bessel beams, Airy beams, Weber beams, and Bessel beams.

[0098] Referring to FIGS. 1A and 1B, each beam 122, 124, 126 of a laser beam combination120 used to form the defects has an intensity distribution I(X,Y,Z), where Z is the beam propagation direction of a particular beam 122, 124, 126, and X and Y are directions orthogonal to the direction of propagation, as depicted in the figures. Indeed, the beam propagation direction of the first beam 122 is Z1, the beam propagation direction of the second beam 124 is Z2, and the beam propagation direction of the third beam 126 is Z3. Planes orthogonal to the respective beam propagation directions Z1, Z2, Z3may be referred to as a cross-sectional plane and the intensity distribution of each beam 122, 124, 126 in a cross-sectional plane may be referred to as a cross- sectional intensity distribution. FIG.1A depicts a perspective view of the beams 122, 124, 126 as they are incident on an impingement surface 162 of a transparent workpiece 160. FIG.1B depicts a cross-sectional view of the transparent workpiece 160 through a defect plane 180. The deflect plane 180 may represent a plane containing the beams 122, 124, 126 when the beams 122, 124, 126 form defects in the transparent workpiece. In the depicted embodiment, the defect plane 180 extends perpendicular to the impingement surface 162 through the thickness of the transparent workpiece 160 and parallel to an external edge of the transparent workpiece 160, though other orientations of the defect plane are contemplated and within the scope of the present disclosure.Attorney Docket No.: SP24-029_PCT

[0099] Each beam 122, 124, 126 at a beam spot 112A, 112B, 112C or other cross section maycomprise a quasi-non-diffracting beam, for example, a beam having low beam divergence as mathematically defined below, by transmitting the initial laser beam 12 through both the first refractive optical element 144A and the second refractive optical element 144B, as described in more detail below with respect to the optical assembly 100 depicted in FIG.3. Beam divergence refers to the rate of enlargement of the beam cross section in the direction of beam propagation (i.e., the Z1, Z2, Z3directions). As used herein, the phrase “beam cross section” refers to the cross section of a beam (e.g., beams 122, 124, 126) along a plane perpendicular to the beam propagation direction Z1, Z2, Z3 of the that respective beam.

[0100] The length of the laser beam focal line produced from a quasi-non-diffracting beam isdetermined by the Rayleigh range of the quasi-non-diffracting beam. Particularly, the quasi-non- diffracting beam defines a laser beam focal line (e.g., 125A, 125B, 125C) having a first end point and a second end point each defined by locations where the quasi-non-diffracting beam has propagated a distance from the beam waist equal to a Rayleigh range of the quasi-non-diffracting beam. The length of the laser beam focal line corresponds to twice the Rayleigh range of the quasi- non-diffracting beam. A detailed description of the formation of quasi-non-diffracting beams and determining their length, including a generalization of the description of such beams to asymmetric (such as non-axisymmetric) beam cross sectional profiles, is provided in U.S. Pat. No.10,730,783 which is incorporated by reference in their entireties.

[0101] The Rayleigh range corresponds to the distance (relative to the position of the beam waistas defined in Section 3.12 of ISO 11146-1:2005(E)) over which the variance of the laser beam doubles (relative to the variance at the position of the beam waist) and is a measure of the divergence of the cross sectional area of the laser beam. The Rayleigh range can also be observed as the distance along the beam axis at which the peak optical intensity observed in a cross sectional profile of the beam decays to one half of its value observed in a cross sectional profile of the beam at the beam waist location (location of maximum intensity). Laser beams with large Rayleigh ranges have low divergence and expand more slowly with distance in the beam propagation direction than laser beams with small Rayleigh ranges.Attorney Docket No.: SP24-029_PCT

[0102] Beam cross section is characterized by shape and dimensions. The dimensions of thebeam cross section are characterized by a spot size of the beam. For a Gaussian beam, spot size is frequently defined as the radial extent at which the intensity of the beam decreases to 1 / e2of itsmaximum value. The maximum intensity of a Gaussian beam occurs at the center (^^ = 0 and ^^ =0 (Cartesian) or ^^ = 0 (cylindrical)) of the intensity distribution and radial extent used to determinespot size is measured relative to the center.

[0103] Beams with Gaussian intensity profiles may be less preferred for laser processing to formdefect segments 172A, 172B, 172C because, when focused to small enough spot sizes (such as spot sizes in the range of microns, such as about 1-5 µm or about 1-10 µm) to enable available laser pulse energies to modify materials such as glass, they are highly diffracting and diverge significantly over short propagation distances (low Rayleigh range). To achieve low divergence (high Rayleigh range), it is desirable to control or optimize the intensity distribution of the pulsed laser beam to reduce diffraction. Pulsed laser beams may be non-diffracting or weakly diffracting. Weakly diffracting laser beams include quasi-non-diffracting laser beams. Representative weakly diffracting laser beams include Bessel beams, Gauss-Bessel beams, Airy beams, Weber beams, and Mathieu beams.

[0104] Non-diffracting or quasi-non-diffracting beams generally have complicated intensityprofiles, such as those that decrease non-monotonically vs. radius. By analogy to a Gaussian beam,an effective spot size ^^^^,^^^^^^ can be defined for any beam, even non-axisymmetric beams, as theshortest radial distance, in any direction, from the radial position of the maximum intensity (r = 0) at which the intensity decreases to 1 / e2of the maximum intensity. Further, for axisymmetric beams ^^^^,^^^^^^is the radial distance from the radial position of the maximum intensity (r = 0) at which the intensity decreases to 1 / e2of the maximum intensity. A criterion for Rayleigh range ZRbased on the effective spot size^^^^,^^^^^^for axisymmetric beams can be specified as non-diffracting or quasi- non-diffracting beams for forming damage regions in Equation (1), below:where ^^^^ is a dimensionless divergence factor having a value of at least 10, at least 50, at least100, at least 250, at least 500, at least 1000, in the range from 10 to 2000, in the range from 50 toAttorney Docket No.: SP24-029_PCT 1500, in the range from 100 to 1000. For a non-diffracting or quasi-non-diffracting beam thedistance (Rayleigh range), ^^^^ in Equation (1), over which the effective spot size doubles, is ^^^^times the distance expected if a standard Gaussian beam profile were used. The dimensionlessdivergence factor ^^^^ provides a criterion for determining whether or not a laser beam is quasi-non-diffracting. As used herein, the first, second, and third beams 122, 124, 126 are considered quasi-non-diffracting if the characteristics of the laser beam satisfy Equation (1) with a value of ^^^^ ^ 10.As the value of ^^^^ increases, the first, second, and third beams 122, 124, 126 approach a morenearly perfectly non-diffracting state.

[0105] Additional information about Rayleigh range, beam divergence, intensity distribution,axisymmetric and non-axisymmetric beams, and spot size as used herein can also be found in the international standards ISO 11146-1:2005(E) entitled “Lasers and laser-related equipment—Test methods for laser beam widths, divergence angles and beam propagation ratios—Part 1: Stigmatic and simple astigmatic beams”, ISO 11146-2:2005(E) entitled “Lasers and laser-related equipment—Test methods for laser beam widths, divergence angles and beam propagation ratios—Part 2: General astigmatic beams”, and ISO 11146-3:2004(E) entitled “Lasers and laser- related equipment—Test methods for laser beam widths, divergence angles and beam propagation ratios—Part 3: Intrinsic and geometrical laser beam classification, propagation and details of test methods”, the disclosures of which are incorporated herein by reference in their entirety.

[0106] Referring now to FIGS. 2A and 2B, a two-dimensional representation of the formationof a quasi-non diffracting beam (e.g., a Bessel beam) via an axicon 240 is depicted. FIGS.2A and 2B are illustrative of the formation of the laser beam focal lines described herein. In FIGS. 2A and 2B, a Gaussian beam is incident on a first (upstream) side 242 of the axicon 240 and rays 210 are shown focusing into a laser beam focal line 212 on the second (downstream) side 244 of the axicon 240. Rays 210 leaving the axicon 240 at a radius, R, will intersect on the laser beam focal line 212 at a corresponding Z location, forming a cone of light comprising a cone-angle θcone. FIG. 2A depicts the production of a full Bessel beam by filling the clear aperture of the axicon 240 with an initial (e.g., input) beam and FIG.2B depicts the production of a shortened Bessel beam (e.g., the laser beam focal line 212) with active focal region in the range:Attorney Docket No.: SP24-029_PCT

[0107] where ^^0 < ^^ < ^^^^ is made by restricting the initial beam to an annular slice with ^^0 <^^ < ^^^^. Indeed, FIG. 2A shows how the laser beam focal line 212 depends on both the radialextent of the input beam and the cone-angle (θcone) of the rays 210. The transformation from R to Z is approximately given by Equation 4: ^^ = ^^ൗ tan^^^^^^^^^^

[0108] and the length of the laser beam focal line 212 is shown by Equation (5):

[0109] Laser beam focal lines 212 having a length of a few mm can be made with relativelylarge input laser diameters (and therefore have a large Rmax). However, the intensity of the laser beam focal line 212 is inversely proportional to its length. FIG.2B shows how an annular slice of the incoming beam (R0<R<Rf) will focus into the laser beam focal line 212 over the focal region defined by Z0<Z<Zf. The power of the laser beam focal line 212 over the distance from Z0to Zfproduced by an annular slice of the input beam extending from R0 to Rf is equal to the integrated power contained in the annular slice – simply the area of the slice multiplied by the integrated beam intensity in that region. Since the area of an annular slice is directly proportional to the difference of squares of R0 and Rf , slices originating close to the center of the axicon will contain less power than those originating from larger radii.

[0110] Referring again to FIGS. 1A and 1B, an example transparent workpiece 160 isschematically depicted undergoing laser processing according to the methods described herein. In particular, FIGS. 1A and 1B schematically depict directing the laser beam combination 120 comprising the first beam 122, the second beam 124, and the third beam 126 simultaneously into the transparent workpiece 160. While the laser beam combination 120 described herein includes three beams, it should be understood that the methods described herein are applicable to laser beam combinations comprising just two beams and laser beam combinations comprising more than threeAttorney Docket No.: SP24-029_PCT beams. As depicted in FIGS.1A and 1B, the first beam 122, the second beam 124, and the third beam 126 are oriented along respective first, second, and third beam propagation axes 121A, 121B, and 121C. The first beam 122 is oriented along the first beam propagation axis 121A and directed into the transparent workpiece 160 at a first impingement location 111A with a first beam propagation angle θbp1. The second beam 124 is oriented along the second beam propagation axis 121B and directed into the transparent workpiece 160 at a second impingement location 111B with a second beam propagation angle θbp2. The third beam 126 is oriented along the third beam propagation axis 121C and directed into the transparent workpiece 160 at a third impingement location 111C and a third beam propagation angle θbp3. The beam propagation angle θbp of each of the first, second, and third beams 122, 124, 126 comprises the average angle of light rays of the particular beam 122, 124, 126 impinging the impingement surface 162 relative to a plane orthogonal to the impingement surface 162 at each beam’s respective impingement location (as shown in FIG.1B).

[0111] The first impingement location 111A, the second impingement location 111B, and thethird impingement location 111C are each laterally offset from one another along the impingement surface 162 of the transparent workpiece 160. For example, the second impingement location 111B is positioned between the first impingement location 111A and the third impingement location 111C. The first beam 122 forms a first beam spot 112A projected onto the impingement surface 162 at the first impingement location 111A. The first beam 122 also forms a first laser beam focal line 125A in the transparent workpiece 160 and generates an induced absorption to produce a first defect segment 172A within the transparent workpiece 160. The second beam 124 forms a second beam spot 112B projected onto the impingement surface 162 at the second impingement location 111B. The second beam 124 also forms a second laser beam focal line 125B in the transparent workpiece 160 and generates an induced absorption to produce a second defect segment 172B within the transparent workpiece 160. The third beam 126 forms a third beam spot 112C projected onto the impingement surface 162 at the third impingement location 111C. The third beam 126 also forms a third laser beam focal line 125C in the transparent workpiece 160 and generates an induced absorption to produce a third defect segment 172C within the transparent workpiece 160. The first, second, and third impingement locations 111A, 111B, 111C are specific locations on the impingement surface 162 where the first, second, and third beams 122, 124, 126 respectively, are first incident upon and initially contact the impingement surface 162. When atAttorney Docket No.: SP24-029_PCT least one of the laser beam combination 120 (including the first, second, and third beams 122, 124, 126 and first, second and third laser beam focal lines 125A, 125B, 125C) and the transparent workpiece 160 are translated relative to one another, the impingement locations 111A, 111B, 111C change.

[0112] A phase alteration is applied to each of the first, second, and third beams 122, 124, 126such that the first, second, and third laser beam focal lines 125A, 125B, 125C each comprise a circular angular spectrum within the transparent workpiece 160 and exhibit a quasi-non-diffracting character (as mathematically defined above in Eq. (1)) within the transparent workpiece 160. In the embodiments depicted in FIG. 1A and 1B, the first laser beam focal line 125A comprises a chamfer angle θCH1 of greater than 5° relative to a plane 106A orthogonal to the impingement surface 162 at the first impingement location 111A (in either a clockwise or counterclockwise rotational direction with respect to the plane 106A), the second laser beam focal line 125B comprises a chamfer angle θCH2 of less than 1° (such as 0°) relative to a plane 106B orthogonal to the impingement surface 162 at the second impingement location 111B (in either a clockwise or counterclockwise rotational direction with respect to the plane 106B), and the third laser beam focal line 125C comprises a chamfer angle θCH3 of greater than 5° relative to a plane 106C orthogonal to the second surface 164 at the exit location 118 (in a rotational direction opposite the rotational direction of the chamfer angle θCH1). The chamfer angle θCH1 and the chamfer angle θCH3 may be from 5° to 40°, such as 8° to 35°, 10° to 40°, 20° to 40°, or the like, for example, 6°, 7°, 8°, 9°, 10°, 11°, 12°, 13°, 14°, 15°, 16°, 17°, 18°, 19°, 20°, 21°, 22°, 23°, 24°, 25°, 26°, 27°, 28°, 29°, 30°, 31°, 32°, 33°, 34°, 35°, 36°, 37°, 38°, 39°, or any range having any two of these values as endpoints.

[0113] It should be understood that the above chamfer angles are example chamfer angles of thelaser beam focal lines 125A, 125B, 125C and that other chamfer angles are contemplated. Indeed, as another example, the second laser beam focal line 125B may comprise a chamfer angle θCH2greater than 5° relative to the plane 106B orthogonal to the impingement surface 162 at the second impingement location 111B. Moreover, because the first, second, and third laser beam focal lines 125A, 125B, 125C comprise chamfer angles θCH1, θCH2, θCH3, respectively, the resultant defect segments 172A, 172B, 172C formed by induced absorption comprise defect angles θd1,θd1,θd1, that are equal to or about equal to the respective chamfer angles θCH1, θCH2, θCH3. Thus, some or all ofAttorney Docket No.: SP24-029_PCT the defect segments 172A, 172B, 172C may be angled, where “angled” refers to an angular deviation from the direction normal to the impingement surface 162 at the respective impingement locations 111A, 111B, 111C or second surface 164 at exit location 118.

[0114] Moreover, each laser beam focal line 125A, 125B, 125C may further include a pluralityof rays. Each individual ray of each laser beam focal line 125A, 125B, 125C may have the samephase, ^^, when converging to form a circular angular spectrum within the transparent workpiece160. Each laser beam focal line 125A, 125B, 125C may have a length in a range of from 0.01 mm to 10 mm or in a range of from 0.1 mm to 5 mm. Various embodiments may be configured to have laser beam focal lines 125A, 125B, 125C with a length l of 0.01 mm, 0.05 mm, 0.1 mm, 0.2 mm, 0.5 mm, 0.7 mm, 1 mm, 2 mm, 3 mm, 4 mm, or 5 mm, or any length between such values. Moreover, the rays of the laser beam focal lines 125A, 125B, 125C comprise an intensity that may be great enough to damage the glass of the transparent workpiece 160. Without intending to be limited by theory, an intensity great enough to damage the glass may correspond to the burst number, the type of glass, the focal spot size, and the pulse width, among other variables. As an example, and not by way of limitation, the damage threshold for glass may be from 1 to 2 terawatts per square centimeter (TW / cm2). In embodiments, each laser beam focal line 125A, 125B, 125C may have an intensity of greater than 4 TW / cm2, 5 TW / cm2, or 6 TW / cm2. In embodiments, the intensity may be greater than the damage threshold and only cause Type 1 damage. Type 1 damage changes the refractive index of the glass without cracking the glass.

[0115] Without intending to be limited by theory, the rays of the first, second, and third beams122, 124, 126 refract upon entry into the transparent workpiece 160. Due to this refraction, the first and third beam propagation angles θbp1,θbp3are greater, in absolute terms (i.e., more positive or more negative), than the first and the third chamfer angles θCH1,θCH3,respectively. Moreover, when the second beam 124 impinges the impingement surface 162 orthogonal the impingement surface 162, the second beam propagation angle θbp2 is equal to the second chamfer angle θCH2. It should be understood that, in such embodiments, refraction of individual rays of the second beam 124 does occur upon entry into the transparent workpiece 160 at normal incidence but, due to the orthogonal geometry, this refraction does not alter the second chamfer angle θCH2 with respect to the second beam propagation angle θbp2.Attorney Docket No.: SP24-029_PCT

[0116] In operation, the laser processing depicted in FIGS. 1A and 1B further includestranslating at least one of the laser beam combination 120 (i.e., the laser beam focal lines 125A, 125B, 125C) and the transparent workpiece 160 relative to each other in a translation direction 101 along a contour line 165 (i.e., a line of desired separation) to form a plurality of defects 172 comprising defect segments 172A, 172B, 172C. The plurality of defects 172 form a contour 170 which may be used to separate the transparent workpiece 160 into a plurality of separated articles 260' (FIG. 12B). The defects 172 may extend, for example, partially or completely through the depth (i.e., the thickness) of the transparent workpiece 160. Furthermore, each of the defects 172 may comprise a C-chamfer shape in which the first and third defect segments 172A, 172C are angled at mirrored angles with respect to a plane orthogonal to the impingement surface 162 of the transparent workpiece 160 and the second defect segment 172B is orthogonal the impingement surface 162 of the transparent workpiece 160.

[0117] Referring now to FIG. 1B, the first laser beam focal line 125A and the first defectsegment 172A extend between the impingement surface 162 and a first termination depth 115 within the transparent workpiece 160 and each terminate at a first termination location 114 which is positioned at the first termination depth 115. In some embodiments, the first laser beam focal line 125A and the first defect segment 172A extend from the impingement surface 162 to the first termination depth 115. However, in other embodiments, the first laser beam focal line 125A and the first defect segment 172A may begin within the transparent workpiece below the impingement surface 162 and termination at the first termination depth 115. As used throughout this disclosure, the term “termination point” refers to the termination point of the region of induced absorption for at least a portion of a laser beam focal line. The “termination point” as defined herein, may ultimately be the termination point for at least a portion of a defect segment within the transparent workpiece 160.

[0118] The second laser beam focal line 125B and the second defect segment 172B extend fromthe first termination location 114 to a second termination location 116, which is positioned at a second termination depth 117 within the transparent workpiece 160. As shown in FIG. 1B, the first termination depth 115 is closer to the impingement surface 162 than the second termination depth 117. Furthermore, the third laser beam focal line 125C and the third defect segment 172C extend from the second termination location 116 toward a second surface 164 of the transparentAttorney Docket No.: SP24-029_PCT workpiece 160, which is opposite the impingement surface 162. In some embodiments, the third laser beam focal line 125C and the third defect segment 172C terminate within the transparent workpiece 160 between the second termination location 116 and the second surface 164. In other embodiments, the third laser beam focal line 125C and the third defect segment 172C reaches the second surface at an exit location 118. Moreover, in some embodiments, the exit location 118 is positioned along the plane 106A orthogonal to the impingement surface 162 at the first impingement location 111A. For example, in embodiments in which the first and third chamfer angles θCH1, θCH3 are mirrored, the second chamfer angle θCH2 is 0° and the distance from the impingement surface 162 and the first termination depth 115 is equal to the distance from the second surface 164 and the second termination depth 117. This forms a C-chamfered edge when separated, as shown in FIGS. 16A and 16B by separated article 260′, which comprises a C- chamfered edge 268.

[0119] Referring now to FIG. 3, an optical assembly 100 for producing the laser beamcombination 120 is schematically depicted. The optical assembly 100 comprises a beam source 10, such as a Gaussian beam source, that outputs an initial laser beam 12, which may comprise a Gaussian beam. The beam source 10 may comprise any known or yet to be developed beam source 10 configured to output laser beams, for example, pulsed laser beams or continuous wave laser beams. In some embodiments, the beam source 10 may output an initial laser beam 12 comprising a wavelength of, for example, 1064 nm, 1030 nm, 800 nm, 532 nm, 515 nm, 400 nm, 355 nm, 343 nm, or 266 nm, or 257 nm. The initial laser beam 12 used to form defects 172 in the transparent workpiece 160 may be well suited for materials that are transparent to the selected laser wavelength and the transparent workpiece 160 may be positioned such that initial laser beam 12 output by the beam source 10 is phase altered by the first and second refractive optical elements 144A and 144B and thereafter irradiates the transparent workpiece 160 as the laser beam combination 120, for example, after impinging the first and second refractive optical elements 144A and 144B and thereafter, the lens assembly 130. Further, a beam pathway 110 may extend from the beam source 10 to the transparent workpiece 160 such that when the beam source 10 outputs the initial laser beam 12, the initial laser beam 12 and (after phase alteration) the laser beam combination 120 traverses (or propagates along) the beam pathway 110.Attorney Docket No.: SP24-029_PCT

[0120] Referring still to FIG. 3, the optical assembly 100 comprises a first refractive opticalelement 144A and a second refractive optical element 144B. The first and second refractive optical elements 144A and 144B form an optical system 140 for forming the laser beam combination 120. In operation, the first and second refractive optical elements 144A and 144B first phase alter the initial laser beam 12, via the first refractive optical element 144A, to split the initial laser beam 12 into the first, second, and third beams 22, 24, 26 (the first, second and third beams 22, 24, 26 are equivalently called the “plurality of beamlets” or “first, second, and third beamlets 22, 24, 26” herein) and thereafter, via the second refractive optical element 144B, phase alter the plurality of beamlets with at least a quasi-non-diffracting phase, which is a phase that generates the laser beam combination 120 comprising the quasi-non-diffracting laser beams 122, 124, 126, such that the first, second, and third beams 122, 124, 126 form first, second, and third laser beam focal lines 125A, 125B, 125C in the transparent workpiece 160 (the quasi-nondiffracting beams 122, 124, 126 are equivalently referred to as the “beams 122, 124, 126” herein). Moreover, while a single initial laser beam 12 is depicted in FIG.3, it should be understood that additional input beams may be used to form the laser beam combination 120. For example, three initial beams may be output by the beam source 10 or from multiple beam sources, such as a first beam source, a second beam source, and a third beam source. In such embodiments, the first refractive optical element 144A may be omitted.

[0121] The first and second refractive optical elements 144A and 144B are positioned within abeam pathway 110 between the beam source 10 and the transparent workpiece 160, in particular, between the beam source 10 and the lens assembly 130 such that the initial laser beam 12 impinges the first and second refractive optical elements 144A and 144B to generate the first, second and third beamlets 22, 24, 26 via the first refractive optical element 144A and convert the plurality of beamlets into quasi-non-diffracting laser beams 122, 124, 126 via the second refractive optical element 144B that are relayed to the transparent workpiece via the lens assembly 130. The first refractive optical element 144A imparts a first phase distribution to the initial laser beam 12 to divide the initial laser beam 12 into the plurality of beamlets, with each beamlet propagating along a separate propagation axis. The second refractive optical element 144B comprises a plurality of beamlet regions, with each beamlet region being configured to apply a different phase distribution to a separate one of the plurality of beamlets so that each of the first, second, and third beams 122,Attorney Docket No.: SP24-029_PCT 124, 126 is quasi-non-diffracting downstream the first and second refractive optical elements 144A and 144B.

[0122] A process for determining the phase distributions (or “phase masks”) to be imparted bythe first and second refractive optical elements 144A and 144B will now be described. The process begins by determining a desired shape for the defect segments 172A, 172B and 172C. As described in greater detail herein with respect to FIG.14, the lengths of each of the defect segments 172A, 172B, and 172C, as well as the chamfer angles θCH1, θCH2, θCH13determine the necessary beam propagation angles θbp1,θbp2,θbp3(see FIG 1B) and size of the first, second, and third beams 122, 124, 126, which determines the necessary impingement locations 111A, 111B, 111C for the first, second, and third propagation axes 121A, 121B, and 121C. That is, a desired edge construction for the component being produced via laser processing of the transparent workpiece 160 determines the lengths and orientations of the defect segments 172A, 172B, 172C.

[0123] The phase distributions needed to be applied to the light from the initial laser beam 12by the first and second refractive optical elements 144A and 144B will depend on the remaining components of the optical assembly 100. For example, as depicted in FIG. 3, the lens assembly 130 relays the first, second, and third beamlets 122, 124, 126 so that the first, second, and third focal lines 125A, 125B, 125C are formed within the transparent workpiece 160. As such, the configuration of the lens assembly 130 (i.e., including a first lens 131 and a second lens 132 as shown in FIG.3) will determine the positioning and propagation directions of the laser beams 122, 124, 126 needed upstream of the lens assembly 130 to provide the defects in a desired arrangement. After the configuration of the lens assembly 130 is identified, modeling can be used to determine phase distributions to be imparted by the first and second refractive optical elements 144A and 144B. In embodiments, a combination of beam propagation models and machine learning algorithms are used to calculate the necessary phases need at each of the first and second refractive optical elements 144A and 144B. In an example, a first beam propagation model is used todetermine the ideal amplitude and ideal phase of the first, second, and third beams 122, 124, 126(upon exiting the second refractive optical element 144B) which can form a chamfer-shaped intensity distribution, given the construction of the lens assembly 130 and positioning of the transparent workpiece 160. In embodiments, the ideal phase comprises an aberration-corrected quasi-non-diffracting phase and a linear (prism) phase. The aberration-corrected quasi-non-Attorney Docket No.: SP24-029_PCT diffracting phase comprises an oblong angular spectrum, described in international patent application publication no. WO 2021 / 158458 A1, hereby incorporated by reference in its entirety, so that the beams entering the transparent workpiece 160 at a non-zero incidence angle (the first beam 122 and the third beam 122 in this example) have an oblong angular spectrum prior to being incident on the transparent workpiece 160 and a circular angular spectrum within the transparent workpiece 160. Such a phase correction is also described in international patent application publication no. WO 2023 / 009331 A1. In addition, the ideal phase can also include phases configured to compensate for other components in the assembly 100. For example, aberrations introduced into the first, second, and third beams 122, 124, and 126 by the first and second lenses 132, 134 could be measured or modelled, and corresponding compensation phases could be added to the ideal phase.

[0124] After the first beam propagation model is used to determine the ideal phases andamplitudes just upstream of the second refractive optical element 144B, a machine learning algorithm is used to determine a first phase mask for the first refractive optical element 144A that can realize the previously calculated ideal amplitude. The first phase mask may be configured to provide the ideal amplitude at an upstream surface 148 of the second refractive optical element 144B so that the second phase mask implemented by the second refractive optical element 144B applies the ideal phases. In embodiments, Fourier transform-based numerical phase-shaping methods, such as the Gerchberg-Saxton (“GS”) algorithm, are used to first phase masks. In such embodiments, a Fourier transform lens (not depicted) can be employed in the optical assembly 100 between the first and second refractive optical elements 144A and 144B. In embodiments without such a Fourier transform lens, another machine learning algorithm, such as adjacent sensitivity analysis (“AA") can be employed.

[0125] After the first phase mask is calculated, a second beam propagation model is used todetermine the phase accumulated by the initial laser beam 12 after being shaped via the first phase mask (implemented by the first refractive optical element 144A). The second beam propagation model is used to determine the phase accumulated by the initial laser beam 12 during propagation between the first refractive optical element 144A and the second refractive optical element 144B. A second phase mask, to be implemented by the second refractive optical element 144B, is then calculated by subtracting the accumulated phase from the previously calculated ideal phase.Attorney Docket No.: SP24-029_PCT

[0126] It has been found that, if the GS or AA algorithms are left unconstrained in the previouslydescribed calculation of the first and second phase masks to be implemented by the first and second refractive optical elements 144A and 144B described herein, that the resultant phase masks include rapid and randomly oriented changes in phase. FIGS.4A-4D are example phase masks calculated using such methods in accordance with an example. The colorbar shows wrapped phase in radians from -π to π and the X and Y scales are in pixels (9.2 µm / px). FIGS.4A-4B depict example first and second phase masks computed with the GS algorithm, respectively. FIGS. 4C-4D depict example first and second phase masks computed with the AA algorithm. Phases were wrapped from -π to π (see legend). As shown, each of the exemplary phase masks have sections where the phase changes from a minimal value to a maximal value over relatively short distances. Moreover, many regions of constant phase have irregular shapes (e.g., boundaries with discontinuous shapes). Such irregular surface shapes have been found to generally be incompatible with current existing grinding and polishing techniques used to form refractive optics.

[0127] Accordingly, various constraints were applied to the GS and AA algorithms with theobject of providing more smoothly varying phase slopes to enable refractive optic fabrication with subtractive machining methods. To effectively apply constraints and ensure model convergence, a particular design for the PM1 mask was selected. Particularly, an approach was used where the initial laser beam 12 (see FIG. 3) was divided into different spatial portions that is incident on a different individual sub-mask, with each sub-mask forming one of the first, second, and third beamlets 22, 24, 26. In an example, a design where the first refractive optical element 144A (see FIG.3) includes a plurality of concentric regions configured to separate different spatial portions of the initial laser beam 12 into the plurality of beamlets was used.

[0128] FIG. 5A schematically depicts a plurality of concentric regions 200 for the first refractiveoptical element 144A, according to an example embodiment. In embodiments, the plurality of concentric regions 200 comprises a peripheral shape corresponding to a portion of the initial laser beam 12 intended to be formed into one of the first, second, and third beamlets 22, 24, 26. For example, in embodiments, the initial laser beam 122 is an axisymmetric laser beam (e.g., a Gaussian laser beam), and so the plurality of concentric regions 200 comprises a circular outer peripheral shape. In the depicted example, the plurality of concentric regions comprises a central region 200A configured to generate the first beamlet 22 from a central portion of the initial laserAttorney Docket No.: SP24-029_PCT beam 12, a first annular region 200B configured to generate the second beamlet 24 from a first annular portion of the initial laser beam 12 circumferentially surrounding the central portion, and a second annular region 200C configured to generate the third beamlet 26 from a second annular portion of the laser beam circumferentially surrounding the first annular portion. The shapes of each of the plurality of concentric regions 200 are chosen to mimic the shape of the ideal target amplitude (after the second phase mask) as much as possible. The sizes of the concentric regions 200 can be selected so that each of the regions interacts with a portion of the initial laser beam 12 having substantially equal power (i.e., a difference between maximum value and a minimum value of the powers for each portion is less than 10% of the minimum value).

[0129] In embodiments, each of the first, second, and third beamlets 22, 24, 26 has a target shape(i.e., beam cross-sectional intensity distribution) for incidence on the second phase mask that is determined based on the ideal phases and amplitudes initially determined based on the transparent workpiece 160 being processed. The chosen machine learning algorithm (e.g., the AA algorithm or GS algorithm) is run on each individual beamlet so as to convert the particular portion of the laser beam being modified by a particular concentric region to the target shape upon being incident on the second phase mask. The target shape may vary in shape from the portion of the initial laser beam 12 used to form a particular beamlet. For example, the portion of the initial laser beam 12 may be circular, and the target shape for the resultant beamlet could be annular. Alternatively or additionally, the target shape may vary in size from the portion of the initial laser beam 12 used to form a particular beamlet.

[0130] FIG. 5B schematically depicts an illustrative example of a central portion 202A, a firstannular portion 202B, and a second annular portion 202C of an initial laser beam 12. The central portion 202A can be received by the central region 200A described with respect to FIG.5A and have a first target shape 204A. The first annular portion 202B can be received by the first annular region 200B described with respect to FIG.5A and have a second target shape 204B. The second annular portion 202C can be received by the second annular region 200C described with respect to FIG.5A and have a third target shape 204C. As shown, the first target shape 204A differs from the shape of the central portion 202A in that the first target shape 204A has a smaller radius than that of the central portion 202A. The second target shape 204B differs from the shape of the first annular portion 202B in that the second target shape 204B is circular in shape and has a smallerAttorney Docket No.: SP24-029_PCT outer radius. The third target shape 204C differs from the shape of the second annular portion 202C in that the third target shape 204C is circular in shape and has a smaller outer radius. In embodiments, each of the first, second, and third target shapes 204A, 204B, 204C is annular. In such embodiments, each beamlet is a ring after interacting with the first phase mask but transforms to the target annular shape as it propagates to the second phase mask.

[0131] The AA or GS algorithm (as modified herein) is used to calculate phase masks thatconvert the portion of the initial laser beam 12 being processed to the target shape downstream from the first phase mask. As shown, the central region 200A in FIG.5A is configured to apply a phase mask C to the central portion 202A so that the first beamlet 22 has the first target shape 204A upon being incident on the second section 145 (or the second phase mask). The first annular region 200B is configured to apply a phase mask B to the first annular portion 202B so that the second beamlet 24 has the second target shape 204B upon being incident on the second section 145. The second annular region 200C is configured to apply a phase mask A to the second annular portion 202C so that the third beamlet 26 has the third target shape 204C upon being incident on the second section 145.

[0132] In embodiments, the phases calculated by the employed machine learning algorithm (e.g.,GS or AA) do not result in the first, second, and third beams 22, 24, 26 having a desired arrangement on the second phase mask (i.e., if the phases were unmodified, the portions 202A, 202B, 202C would not arrive at the second phase mask at the desired locations). That is, the phase masks output by the GS or AA algorithms may result in an initial phase that does not change a propagation direction of the portion of the laser beam being modified. Accordingly, linear phases can be added to the phases calculated by the algorithm (e.g., the phases A, B, and C described with respect to FIG.5B) so that the first, second, and third beams 22, 24, 26 have a desired arrangement when incident on the second phase mask. The linear phases can act as prism applying a linear varying phase as a function of distance from the optical axis to shift location at which the modified portions of the initial laser beam are incident on the second phase mask. The desired arrangement of the beamlets can vary depending on the desired arrangement of defects within the transparent workpiece 160. In embodiments, the beams are directed to distinct, non-overlapping portions of the second phase mask (corresponding to the “beamlet regions” of the second refractive optical element 144B described herein) that are sized and shaped to correspond to the ideal amplitude mapAttorney Docket No.: SP24-029_PCT initially computed. FIG.5C depicts the total result of a combined phase (the shape changing phase calculated via the machine learning algorithm and linear phase to direct the beamlet to a desired portion of the second phase mask) applied by the resultant first phase mask. The right side of FIG. 5C schematically depicts a desired arrangement 206 of the first, second, and third beams 22, 24, 26 at the second phase mask, according to an example embodiment. In this example, centers of each of the plurality of beamlets form a line in the transverse optical plane (perpendicular to an optical axis of the optical assembly 100). As shown, the first beam 22 has a smallest cross- sectional area and is at a first end of the line. The third beam 26 has a largest cross-sectional area and is disposed at the other end of the line. The second beam 24 has a cross-sectional area between those of the first and third beams 22 and 26 and is disposed between the first beam 22 and the third beam 26. The desired arrangement 206 is exemplary. For example, at least one of the ordering and spacing of the beams can vary. Moreover, nonlinear arrangement of beams are also possible. It should also be appreciated that, while the target shapes 204A, 204B, 204C shown in FIG. 5B are circular in the depicted example, other target shapes (e.g., annular) are also contemplated and within the scope of the present disclosure.

[0133] After computation of the first phase mask, the second phase mask is then used to calculatethe total accumulated phase at the second phase mask, which is then subtracted from the ideal phase to determine the phase values for the second phase mask. To calculate the phases for the first and second phase masks, applying constraints to the machine learning algorithms utilized was found to be necessary. In the case of the GS algorithm, two constraints were applied: (a) the design space was constrained to radial coordinates (r,φ), forcing the converged solutions to have radial symmetry, reducing mask complexity; and (b) a spherical seed phase was used to initiate the algorithm. An advantage of the constraint (a) is the reduction in computation time for solution convergence. If the target amplitude is oval shaped, then an angular stretching function f(φ) is applied to the phase mask to match the target shape. In the constraint (b), the magnitude of the input phase is controlled to achieve higher accuracy solutions toward improved beam shaping efficiency. The spherical nature of the seed encourages smoother variations in phase change than a random seed phase toward improved manufacturability. When the AA algorithm is employed, a penalty function equal to the second derivative of the phase at the second phase mask multiplied by a weighting factor was added to the AA algorithm’s figure of merit. This forces the algorithmAttorney Docket No.: SP24-029_PCT to find solutions with low changes in phase slope which leads to smoother and more easily manufactured surfaces as well as reducing optical alignment sensitivity.

[0134] FIGS. 6A-6D depict phase masks calculated using the constrained algorithms describedherein. The colorbar shows wrapped phase in radians from -π to π and the X and Y scales are in pixels (9.2 µm / px). FIGS.6A-6B depict first and second phase masks 600, 608 computed using a constrained GS algorithm, while FIGS. 6C-6D depict first and second phase masks 616, 618 computing using a constrained AA algorithm. As shown, compared with the unmodified phase masks depicted in FIGS.4A-4D, the phase masks computed using the constrained algorithms have much smoother varying phases. This is demonstrated by FIG.7. The first and second phase masks calculated via the methods described herein can be converted into a surface height map for a refractive optic. Particularly, the machinable height map is determined by unwrapping the phase of the designed phase masks and converting phase to height by dividing the phase by the wavenumber of the light in the refractive optic medium. FIG. 7 depicts a surface profile (height map) of a line drawn down the center of each unwrapped phase (resulting directly from the algorithm) for the first phase masks depicted in FIGS.4A, 4C, 6A, and 6C. As shown, the phases varied for the unmodified algorithms at a higher average spatial frequency than for the constrained algorithms. Both masks made using modified algorithms have smooth slopes except for the transitions between the phase masks associated with the different portions of the input beam (at 175, 210, 505, and 595 px), while the masks made using unmodified algorithms contain many small jumps that would make manufacturing of an optic difficult using conventional subtractive machining methods.

[0135] To further quantify the smoother phase transitions achieved via the constrainedalgorithms described herein, the phase masks depicted in FIGS. 4A-4E and FIGS. 6A-6D were unwrapped using the method described in Herraez et al., “Fast two-dimensional phase-unwrapping algorithm based on sorting by reliability following a noncontinuous path,” Journal Applied Optics, Vol.41, No.35, pp.7437, 2002, hereby incorporated by reference in its entirety. This resulted in some errors causing large slopes, especially in the masks computed with unmodified algorithms. Large slopes were largely ignored due to the histogram bin size in a calculation of the magnitude of the gradient of the unwrapped phase. FIGS. 8A-8D are histograms for change in phase slope as a function of pixel count for the first phase masks depicted in FIGS. 4A, 4C, 6A, and 6C. InAttorney Docket No.: SP24-029_PCT these calculations, each pixel was a 9.2 µm square. The bin size in each of the histograms is 0.01 rad / px2, so the first bin in each histogram represents a change in phase slope under 0.01 rad / px2. FIGS.8A and 8B are histograms for the change in phase slope for the pixels of the masks shown in FIGS.4A and 4C. FIGS.8C and 8D are histograms for the change in phase slope for the masks shown in FIGS. 6A and 6C. As shown, compared to the masks calculated with the unmodified algorithms, the masks calculated with the constrained algorithms exhibited far greater number of pixels associated within the first bin. This indicates that a substantial majority of the phase distribution of the first phase mask calculated via the methods described herein are associated with a very small change in phase slope and are more amenable to subtractive manufacturing techniques used to make refractive optics.

[0136] FIG. 8E is a plot of the cumulative pixel counts as a function of change in phase slopecalculated from the histograms depicted in FIGS.8A-8D. As shown, the phase masks calculated by the modified algorithms described herein each had more than 75% of the pixels associated with a change in phase slope that is less than 0.01 rad / px2. This starkly contrasts with the phase masks generated via the unmodified phase masks, which had less than 1% of pixels at such low change in phase slope. FIG.8E is a plot of cumulative pixel counts as a function of change in phase slope calculated for the second phase masks depicted in FIGS. 4B, 4D, 6B, and 6D. As shown, the second phase masks calculated by the modified algorithms described herein are distinguished from those calculated by the unmodified algorithms in that greater than 50% of the pixels are associated with a change in phase slope that is less than 0.01 rad / px2. Thus, the computational methods described herein result in a calculation phase masks with relatively low changes in phase slope such that at least 50% of areas of phase distributions applied individually by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2. Unless otherwise noted herein, the phase values associated with the phase masks (and changes in phase and changes in phase slope) are calculated assuming a wavelength of light of 1030 nm.

[0137] The relatively low change in phase slope associated with the phase masks describedherein are believed to have an additional advantage of reduced sensitivity to beam misalignments and manufacturing errors, such as unintentional smoothing of surface features. Misalignment or mis-shaping of the refractive optics could result in aberrations in the quasi-non-diffracting beams,Attorney Docket No.: SP24-029_PCT leading to the laser beam focal lines not having an intended length and / or shape within the workpiece. Reduced sensitivity to misalignment and misshaping thus ensures process reliability and achieving the intended result of the laser processing. To quantify these benefits, two simulations were performed on the pairs of phase masks described herein with respect to FIGS. 4A-4D and 6A-6D (masks calculated with the same algorithm were paired with one another). The first simulation examined errors due to the angular misalignment of an input Gaussian laser beam relative to the first phase mask, resulting in spatial misalignment between the formed beamlets and the second phase mask. The misalignment causes the second phase mask to improperly account for the accumulated phase during propagation from the first phase mask to the second phase mask, where the magnitude of the residual phase is proportional to the rate of change of the slope of the phase in this region. The angular misalignment was simulated by diagonally offsetting (at 45° relative to the X or Y axes) the position of the second phase mask by up to 9 pixels (pixel spacing is 9.2 μm in this simulation – 1 pixel is equivalent to a 36 μrad angular misalignment with 250 mm between the phase masks). The results for the GS algorithm included a 250 mm Fourier transform lens between the phase masks. The results of this simulation are shown in FIG. 9A. As shown, the modified machine learning algorithms produce a quasi-non-diffracting beam with higher average focal intensity under angular misalignment than the unmodified algorithms due to the slower-varying phase slopes, causing smaller errors in accounting for accumulated phase via the second phase mask. Here, average focal intensity is found by integrating the intensity through the beam’s desired focal region (i.e., the laser beam focal line).

[0138] The second simulation compared mask performance after convolution with a Gaussianfunction. This can roughly mimic the effect of tooling with a radiused head (i.e., feature smoothing) and is another test of mask sensitivity to perturbations. The average intensity through focal region of one of the beamlets was compared for different widths of the Gaussian convolution kernel. The results are shown in FIG. 9B. As shown, the masks produced by the constrained algorithms achieve higher average focal intensities than generated using conventional machine learning algorithms as the kernel width increases. These results demonstrate that the phase masks described herein are beneficial over existing holographic-type designs by providing less sensitivity to misalignment and feature smoothing during fabrication. Example Fabrication TechniqueAttorney Docket No.: SP24-029_PCT

[0139] Refractive optics can be fabricated from the phase masks calculated by the methodsdescribed herein by generating a suitable toolpath. The toolpath can be calculated from a machinable height map determined by unwrapping the phase of the designed phase masks and converting phase to height by dividing the phase by the wavenumber of the light in the refractive optic medium. FIG.10A depicts an example first height map 1000 for a first phase mask calculated via the methods described herein. FIG. 10B depicts an example second height map 1010 for a second phase mask calculated via the methods described herein. The first and second height maps 1000 and 1010 were computed using the GS algorithm. It has been found that these height maps contain surface discontinuities that create challenges in using a standard computer aided manufacturing package for machining equipment. For example, the first height map 1000 includes a first discontinuity 1002 between first and second annular regions 200B and 200C (see FIG.5A) of the first refractive optical element 144A. The first height map 1000 also includes a second discontinuity 1004 at an outer boundary of the second annular region 200C. The second height map 1010 includes a discontinuity 1012 at the outer edge of the beamlet regions, separating the background plane of the point cloud and the optical surface implementing the second phase mask.

[0140] The discontinuities 1002, 1004, 1012 create issues in the fabrication of the first andsecond refractive optical elements 144A and 144B described herein. For example, the optical surfaces of the first and second refractive optical elements 144A and 144B can be fabricated using the slow slide servo function of a NanoTech Diamond turning lathe. (A fast tool servo accessory can also be used.) The slow slide servo (or fast tool servo) synchronizes the Z axis motion with the spindle to generate free form optics. Ideally, the spindle speed should be high enough for good cutting performance, but the stroking motion of the Z axis should not have high acceleration spikes (e.g., acceleration of the spindle should not exceed 0.05 g). The discontinuities 1002, 1004, 1012 in the height maps 1000 and 1010 could lead to high accelerations, rendering manufacturing of optical surfaces implementing the calculated phase masks impractical.

[0141] The discontinuities 1002 and 1004 in the first phase height map 1000 were addressed bytreating the different sections (the central region 200A, the first annular region 200B, and the second annular region 200C shown in FIG. 5A) separately using “machining range” function in the standard software package with the lathe. A different toolpath was generated for each individual section, which allowed for high enough spindle RPM and acceptable acceleration spikesAttorney Docket No.: SP24-029_PCT by avoiding the discontinuities. This is an added benefit of imposing rotational symmetry on the different sections of the first phase mask described herein.

[0142] There is no way of segmenting the second height map 1010 since the optical surfacecannot be separated into circular sections. Additionally, the discontinuity 1012 creates issues because generating a tool path directly from the cartesian point-cloud will produce high accelerations on the discontinuities, which makes the tool path not practical. To circumvent this issue, the background of the point cloud was removed and a polar grid was overlayed to the cartesian point cloud. The Z height values where transferred to the polar grid point cloud. To do this, a python script was written that found the closest points in the X-Y plane and then asigned the height value of the cartesian grid to the polar grid. This resulted in height values within the beamlet regions being transferred to outside of the beamlet regions, removing the discontinuity 1012. The polar grid point cloud overlaid onto the cartesion point cloud with the cartesian background removed is shown in FIG. 11. A further benefit of such a point cloud is that the toolpath is now circular. It is important to note that the resolution (how close the points are) for an ordered polar grid is reduced as distance from the center increases. The polar grid density can be adjusted as needed to avoid losing detail on the outer portions of the grid. Once the polar grid is generated it is imported into the standard CAM package for toolpath formation.

[0143] Using these techniques, example first and second refractive optical elements 144A and144B were machined in acrylic using the slow slide servo option of the Nanotech diamond turning machine. Acrylic was chosen as a material for prototyping due to being easily machinable; however, a harder material such as fused silica, barium fluoride, or calcium fluoride could be used to fabricate an optic usable for laser processing. Each refractive optical element included a flat surface and a variable surface to provide thickness variations to implement the spatially varying phase distributions associated with the first and second phase masks. Surface height profiles were taken on the variable surfaces of each of the first and second refractive optical elements via optical profilometry. FIG.12 is a surface height profile of a variable surface 1200 of the first refractive optical element 144A. FIG.13 is a surface height profile of a variable surface 1300 of the second refractive optical element 144B. In each of the profiles shown in FIGS. 12-13, the zoom of the plots is enhanced in the Z-direction to show the geometry of the variable surfaces 1200 and 1300.Attorney Docket No.: SP24-029_PCT The rectangular features shown, particularly for the variable surface 1300, are stitching artifacts from the imaging configuration.

[0144] The variable surfaces 1200 and 1300 will now be described in relation to the opticalassembly 100 described herein with respect to FIG. 3 and FIGS. 5A-5C to provide further detail on the operation of the first and second refractive optical elements 144A and 144B. As a result of the variable surface 1200 depicted in FIG.12, the first refractive optical element 144A according to this example comprises a plurality of concentric regions 1202 configured to separate different spatial portions of the initial laser beam 12 into the plurality of beamlets. In embodiments, the first refractive optical element 144A is centered with respect to the laser beam source 10 so that a propagation axis (corresponding to the optical axis of the optical assembly 100) of the initial laser beam 12 impinges on a geometric center of the variable surface 1200. As shown in FIG.12, the plurality of concentric regions 1202 includes a circular central region 1202A, a first annular region 1202B, and a second annular region 1202C. In the depicted example, the central region 1202A corresponds with the central region 200A described herein with respect to FIG.5 and is positioned to receive the central portion 202A of the initial laser beam 12. The first annular region 1202B corresponds with the first annular region 200B described with respect to FIG.5 and is positioned to receive the first annular portion 202B of the initial laser beam 12. The second annular region 1202 corresponds with the second annular region 200C described with respect to FIG. 5 and is positioned to receive the second annular portion 202C of the initial laser berm 12.

[0145] Each of the plurality of concentric regions 1202 depicted in FIG. 12 can include surfaceheight variations so that the first refractive optical element 144A implements any of the first phase masks described herein. For example, the central region 1202A can have surface height variations so as to implement a central region like a central region 602 of the phase mask 600 shown in FIG. 6A. As described herein, the central region 602 can impart a beam shaping phase onto the central portion 202A so that the central portion 202A possesses the first target shape 204A when incident on the upstream surface 148 of the second refractive optical element 144B. The central region 602 can also impart a linear (prism) phase onto the central portion 202A so that the central portion 202A forms the first beam 22 and is incident on the upstream surface 148 in accordance with the desired arrangement 206 depicted in FIG. 5C. The first annular portion 1202 can have surface height variations to implement an annular region like a first annular region 604 of the phase maskAttorney Docket No.: SP24-029_PCT 600 shown in FIG.6A. As described herein, the first annular region 604 can impart a beam shaping phase onto the first annular portion 202B so that the first annular portion 202B possesses second target shape 204B when incident on an upstream surface 148 of the second refractive optical element 144B. The first annular portion 604 can also impart a linear (prism) phase onto the first annular portion 202B so that first annular portion 202B forms the second target shape 204B and is incident on the upstream surface 148 in accordance with the desired arrangement 206 depicted in FIG. 5C. The second annular region 1202C can have surface height variations to implement an annular region like a second annular region 606 of the phase mask 600 shown in FIG. 6A. As described herein, the second annular region 606 can impart a beam shaping phase onto the second annular portion 202C so that the second annular portion 202C possesses third target shape 204C when incident on an upstream surface 148 of the second refractive optical element 144B. The second annular portion 606 can also impart a linear (prism) phase onto the second annular portion 202C so that second annular portion 202C forms the third target shape 204C and is incident on the upstream surface 148 in accordance with the desired arrangement 206 depicted in FIG. 5C. As such, each of the plurality of concentric regions 1202 applies a different linear phase to a different spatial portion of the initial laser beam 12 so that each of the plurality of beamlets incident on a separate non-overlapping portion of the second refractive optical element 144B in accordance with a desired arrangement computed based on an ideal phase to form the laser beam focal lines 125A, 125B, 125C in the transparent workpiece 160.

[0146] Referring now to FIGS. 3, 5A-5C, and 13, because of the variable surface 1300 shownin FIG. 13, the second refractive optical element comprises a plurality of beamlet regions 1302. In the optical assembly 100 depicted in FIG. 3, the second refractive optical element 144B is positioned relative to the first refractive optical element 144A so that each of the beams 22, 24, 26 generated via the first refractive optical element 144A is incident on a separate one of the plurality of beamlet regions 1302 so that each of the plurality beamlet regions 1302 is configured to apply a different phase distribution to a separate one of beams 22, 24, 26 so that, downstream the second refractive optical element 144B, the beams 122, 124, and 126 all possess a quasi-non-diffracting character. As shown in FIG.13, the variable surface 1300 comprises a first beamlet region 1302A, a second beamlet region 1302B, and a third beamlet region 1302C. Each one of the plurality of beamlet regions 1302 can be sized and positioned based on a particular one of the beams 22, 24, 26 to impart a phase distribution thereto. As shown in FIG.13, geometric centers of each of theAttorney Docket No.: SP24-029_PCT plurality of beamlet regions 1302 can be disposed along a line 1304. In embodiments, the line 1304 is disposed in a transverse optical plane 150 (see FIG. 3) extending perpendicular to the optical axis of the assembly (representing a straight linear optical path from the center of the initial laser beam 12 through the transparent workpiece 160). The line 1304 can also be parallel to the impingement surface 164 in some embodiments to facilitate alignment of the beams 22, 24, 26. Such alignment facilitates the formations of the plurality of beams 122, 124, 126 in a single plane perpendicular to the impingement surface 162 to facilitate formation of defects in the transparent workpiece 160 in a single plane (the “defect plane” described herein).

[0147] Each of plurality of beamlet regions 1302 is sized and positioned based on the desiredarrangement 206 computed based on the ideal phase and amplitude distributions for the beams 22, 24, 26 needed to form a desired arrangement of defects in the transparent workpiece 160. The desired arrangement 206 described herein is a particular example and is not meant to be limiting. The same arrangement of defects in the transparent workpiece could be formed using different sections of the initial laser beam 12 to form different defect segments than those shown in the particular example herein. Additionally, different arrangements of defects (e.g., with different chamfer angles and / or differing number of defect segments), could be accomplished with a different arrangement of beamlet regions than that shown in FIG.13.

[0148] In the example depicted in FIG. 13, the first beamlet region 1302A has the smallestsurface area of the plurality of beamlet regions 1302 and is disposed at a first end of the plurality of beamlet regions 1302 so as to receive the first beam 22. The third beamlet region 1302C has the largest surface area of the plurality of beamlet regions and is disposed at a second end of the plurality of beamlet regions 1302 so as to receive the third beam 26. The second beamlet region 1302B is disposed between the first beamlet region 1302A and the third beamlet region 1302C and has a surface area between those of the first beamlet region 1302A and the third beamlet region 1302C. The surface areas of each of the beamlet regions 1302 can be chosen to closely match the target shapes 204A, 204B, 204C and the portions of the initial laser beam 12 that they receive. Such a construction minimizes the phases needed to induced by the first refractive optical element 144A on the portions of the initial laser beam 12 to provide the desired shapes at the second refractive optical element 144B. The plurality of beamlet regions 1302 can include spatial thickness variations to implement any of the second phases calculated via the methods describedAttorney Docket No.: SP24-029_PCT herein. In an example, the first beamlet region 1302A can implement a beamlet region like the first beamlet region 610 of the phase mask 608 shown in FIG. 6B, the second beamlet region 1302B can implement a beamlet region like the second beamlet region 612 of the phase mask 608, and the third beamlet region 1302C can implement a beamlet region like the third beamlet region 614 of the phase mask 608.

[0149] As described herein, each of the plurality of beamlet regions 1302 is configured to applya corrective phase configured to counteract the phase imparted by one of the plurality of concentric regions 1202 of the first refractive optical element 144A and an aberration corrected quasi-non- diffractive phase to each beamlet. In the depicted example, the first beamlet region 1302A is configured to apply a corrective phase to counteract phases imparted on the central portion 202A by the central region 1302A as well as an aberration quasi-non-diffracting phase to the first beam 22. As a result, the first beam 122 may be quasi-non-diffracting downstream the second refractive optical element 144B and possess an oblong angular spectrum configured to counteract aberrations imparted thereon at the impingement surface 162 as a result of being incident thereon at the first beam propagation angle θbp1. The second beamlet region 1302B is configured to apply a corrective phase to counteract phase imparted on the first annular portion 202B by the first annular region 1202B as well as an aberration quasi-non-diffracting phase to the second beam 24. As a result, the second beam 124 may be quasi-non-diffracting downstream the second refractive optical element 144B and also possess an oblong angular spectrum configured to counteract aberrations imparted thereon at the impingement surface 162 as a result of being incident thereon at the second beam propagation angle θbp2(if θbp2= 0°, the beam 24 may possess a circular angular spectrum as no such aberrations may be introduced at the impingement surface 162). The third beamlet region 1302C is configured to apply a corrective phase to counteract phases imparted on the second annular portion 202C by the second annular region 1302C as well as an aberration quasi-non- diffracting phase to the third beam 26. As a result, the third beam 126 may be quasi-non-diffracting downstream the second refractive optical element 144B and also possess an oblong angular spectrum configured to counteract aberrations imparted thereon at the impingement surface 162 as a result of being incident thereon at the third beam propagation angle θbpc. To ensure that the beams 122, 124, 126 are propagating at desired propagation directions, linear prism phases can also be applied within each beamlet region to ensure that the ideal phase and amplitude distributions are achieved.Attorney Docket No.: SP24-029_PCT

[0150] Referring still to FIG. 13, at least one of the plurality of beamlet regions 1302 can beoriented approximately normal (less than 1° from) the optical axis of the optical assembly 100. In the depicted example, the second beamlet region 1302B is disposed within 1° of normal relative to the optical axis. The orientation of a particular beamlet region relative to the optical axis can indicate a magnitude of a linear prism phase applied to the beam incident on that beamlet region. Since the second beamlet region 1302B is disposed approximately normal to the optical axis, relatively little or no linear phase is applied to the second beam 24 in this example, as the second beam 24 has as a second beam propagation angle θbp2 that is approximately normal to the impingement surface (see FIG. 1B). In embodiments, the first and third beamlet regions 1302A and 1302C are inclined relative to second beamlet region 1302B. The inclination represents the linear prism phases applied to the first and third beams 22 and 26 to ensure propagation in a desired arrangement downstream the second refractive optical element 144B. As shown in FIG. 13, the first beamlet region 1302A is generally inclined relative to the second beamlet region 1302V at a first inclination angle θi1and the third beamlet region 1302C is inclined relative to the second beamlet region 1302B at a second inclination angle θi2. Such inclination angles can be calculated by averaging the surface profile of each beamlet region using a moving area around each point. Inembodiments, the inclination angles θi1, θi2 are determined based on the desired beam propagationangles θbp1, θbp3 of the first and third beams 22, 26 and the magnification of the lens assembly 130.For example, in embodiments, the inclination angles θi1, θi2 are greater than or equal to 0.25° andless than or equal to 2° and the magnification of the lens assembly can vary from 15 to 20, such that beam propagation angles θbp1, θbp3that range from approximately 3.75° to approximately 40° can be provided. As mentioned herein, the Z-axis scale in FIG.13 is exaggerated for the purposes of illustration, so the depicted angles in FIG.13 do not correspond with the actual angles measuredon the variable surface 1300. The inclination angles θi1, θi2 are still present on the variable surface1300 in the preceding ranges.

[0151] In summary, operation of the first and second refractive optical elements 144A and 144Bin the optical assembly 100 can be described as follows. A plurality of spatial regions of the first refractive optical element 144A (the plurality of concentric regions 1300 in the depicted example) are configured such that a first portion of the initial laser beam 12 (the central portion 202A in this example) is directed to the first beamlet region 1302A, a second portion of the initial laser beam 12 (the first annular portion 202B in the depicted example) is directed to the second beamlet regionAttorney Docket No.: SP24-029_PCT 1302B, and a third portion of the initial laser beam 12 (the second annular portion 202C in the depicted example) is directed to the third beamlet region 1302C. Downstream of the second refractive optical element 144B, the plurality of beamlets are quasi-non-diffracting laser beams that are each relayed by the lens assembly 130 into the transparent workpiece 160 to generate defects in segments extending through the transparent workpiece 160 at angles relative to one another in the defect plane. Such a construction is beneficial in that the first and second optical elements can be rotated in conjunction in one another (e.g., the elements can be mounted in a rotatable housing) to rotate the arrangement of the plurality of quasi-non-diffracting laser beams. In the example described herein with respect to FIGS.1A-1C, 3, 5A-6D, 12, and 13 herein, a linear arrangement of quasi-non-diffracting beams can be rotated so as to form a curved contour (e.g., to form a corner of the transparent article being formed). Geometry of Laser Beam Combination for a C-shaped Chamfered Edge

[0152] Referring now to FIG. 14, the geometry of the laser beam combination 120 used to forma separated article 260′ with a C-chamfered edge 268 (FIG. 16B) from a transparent workpiece 160 is schematically depicted. FIG.14 illustrates the spaced-apart nature of the beamlet regions of the second refractive optical element 144B. In FIG. 14, the lens assembly 130 is not shown, but would be placed between first and second refractive optical elements 144A and 144B and transparent workpiece 160.

[0153] Without intending to be limited by theory, the depth location of each of the first, second,and third laser beam focal lines 125A, 125B, 125C within the transparent workpiece 160 is controlled by the inner diameters and the outer diameters of the first, second, and third beams 122, 124, 126 (in some embodiments, the first, second, and third beams are annular in cross-section downstream the second refractive optical element 144B). The radial dimensions of each beam controls the length of the respective laser beam focal lines 125A, 125B, 125C formed from each of the beams 122, 124, 126. The first and second refractive optical elements 144A, 144B, in combination with the lens assembly 130, is used to control both the offset position of each beam 122, 124, 126, the length of each laser beam focal line 125A, 125B, 125C, and the depth location of each laser beam focal line 125A, 125B, 125C.Attorney Docket No.: SP24-029_PCT

[0154] In the embodiment depicted in FIG. 14, the first beam 122, which focuses into the firstlaser beam focal line 125A, forms the top beam of the C-chamfer beam combination of laser beam focal lines 125A, 125B, 125C that form in the transparent workpiece 160. The first beam 122 propagates from a first location 402 of the surface 147 of the second refractive optical element 144B (corresponding to a center of the first beamlet region 1302A shown in FIG.13). The second beam 124, which focuses into the second laser beam focal line 125B, forms the middle beam of the C-chamfer beam combination of laser beam focal lines 125A, 125B, 125C that form in the transparent workpiece 160. The second beam 124 propagates from a second location 404 of the surface 147 (corresponding to a center of the second beamlet region 1302B shown in FIG. 13). The third beam 126, which focuses into the third laser beam focal line 125C, forms the bottom beam of the C-chamfer beam combination of laser beam focal lines 125A, 125B, 125C that form in the transparent workpiece 160. The third beam 126 propagates from a third location 406 of the surface 147 (corresponding to a center of the third beamlet region 1302C shown in FIG.13).

[0155] Without intending to be limited by theory, the lateral offset between the first location 402and the third location 406 depends on the desired starting depth of the first laser beam focal line 125A in the transparent workpiece 160 (i.e., the position of the first laser beam focal line 125A at or nearest the impingement surface 162 of the transparent workpiece 160), the desired ending depth of the third laser beam focal line 125C in the transparent workpiece 160 (i.e., the position of the third laser beam focal line 125C at or nearest the second surface 164 of the transparent workpiece 160), the desired first chamfer angle θCH1, and the desired third chamfer angle θCH3.

[0156] In the embodiment of FIG. 14, axis 400 extends from the surface 147 at an origin location401 between the first location 402 and the third location 406, laterally equidistant from the first location 402 and the third location 406. The origin location 401 is also laterally positioned between the second location 404 and the third location 406. Thus, the second location 404 is nearer the first location 402 than the third location 406. To form a defect that is a symmetric C-chamfer shape, the first beam 122 and the third beam 126 are laterally offset at the surface 147 such that the first beam propagation axis 121A intersects the third beam propagation axis 121C at intersection point 405 location in the center of the transparent workpiece 160. That is, the intersection point 405 is equidistant to the impingement surface 162 and the second surface 164.Attorney Docket No.: SP24-029_PCT

[0157] For a chamfer where ^^^^^^1 = ^^^^^^3 , as depicted in FIG. 14, the lateral offset of the firstlocation 402 and the third location 406 relative to the origin location 401 of the surface 147 are shown approximately by Equations (6) and (7):

[0158] where ^^^^^^^^ is the offset of the first location 402 (i.e., the impingement location of thefirst beam 122 with surface 146 / 147) from the origin location 401, ^^^^^^^^ is the offset of the thirdthe third location 406 from the origin location 401, ^^^^^^^^ is the thickness of the transparentworkpiece 160, ^^ is the refractive index of the transparent workpiece 160, and ^^^^^^^^ is an offsetdistance due to the first beam 122 starting from a nonzero radius. Furthermore, the lateral offset of the second location 404 from the origin location 401 is shown by Equation (8):

[0159] In Equation (8), ^^^^^^^^ is the offset of the second location 404 (i.e., the impingementlocation of the second beam 124 with surface 146 / 147) from the origin location 401, ^^^^^^^^ is thedesired length of the second laser beam focal line 125B within the transparent workpiece 160 (that is, the middle segment of a C-chamfered focal line combination). To form a defect that is asymmetric C-chamfer shape, ^^^^^^^^ may be^^^^^^^^ൗ 3 , i.e., one third of the thickness of the transparentworkpiece 160.

[0160] Referring still to FIG. 14, the depth location within the transparent workpiece 160 andthe length of each laser beam focal lines 125A, 125B, 125C is dependent on the outer diameter of the annulus of each beam 122, 124, 126, the thickness of the annulus of each beam 122, 124, 126(i.e., the difference between the outer diameter and the inner diameter), and the cone angle ^^^^^^^^^^of each beam 122, 124, 126 at the impingement surface 162. Here, the cone angle ^^^^^^^^^^ of eachbeam 122, 124, 126 is the angle from the inner radius of the annulus of each beam 122, 124, 126 to the start of the respective laser beam focal lines 125A, 125B, 125C, the angle from the midpointAttorney Docket No.: SP24-029_PCT radius of the annulus of each beam 122, 124, 126 to the midpoint of the respective laser beam focal lines 125A, 125B, 125C, and the angle from the outer radius of the annulus of each beam 122, 124, 126 to the end of the respective laser beam focal lines 125A, 125B, 125C. Depth location refers to the first termination depth 115 and the second termination depth 117 depicted in FIG. 1B, as well as the starting point of the first laser beam focal line 125A (which may be at the impingement surface 162) and the ending point of the third laser beam focal line 125C (which may be at the second surface 164). The approximate outer diameter of each of the first, second, and third beams 122, 124, 126 at the impingement surface 162 is shown by Equations (9)-(11) :

[0161] In Equation (9), the outer diameter of the first beam 122 at the impingement surface 162is a function of the lateral offset ^^^^^^^^ between the origin location 401 and the first location 402,the desired length of the first laser beam focal line 125Athe refractive index ^^ of thetransparent workpiece 160, and the cone angle ^^^^^^^^^^1 of the first beam 122 at the impingementsurface 162. In Equation (10), the outer diameter of the second beam 124 at the impingementsurface 162 is a function of the lateral offset ^^^^^^^^ between the origin location 401 and the firstlocation 402, the desired length of the first laser beam focal line 125Athe desired lengthof the second laser beam focal line 125B (^^^^^^^^), the refractive index ^^ of the transparent workpiece160, and the cone angle ^^^^^^^^^^2 of the second beam 124 at the impingement surface 162. InEquation (11), the outer diameter of the third beam 126 at the impingement surface 162 is afunction of the lateral offset ^^^^^^^^ between the origin location 401 and the first location 402, thedesired length of the first laser beam focal line 125Athe desired length of the second laser beam focal line 125B (^^^^^^^^), the desired length of the third laser beam focal line 125C (^^^^^^^^), therefractive index ^^ of the transparent workpiece 160, and the cone angle ^^^^^^^^^^3 of the third beam126 at the impingement surface 162. Because the first laser beam focal line 125A forms in the transparent workpiece 160 nearer to the impingement surface 162 than the second laser beam focal line 125B and the second laser beam focal line 125B form in the transparent workpiece nearer theAttorney Docket No.: SP24-029_PCT third laser beam focal line 125C, the outer diameter of the third beam 126 at the impingementsurface 162 (^^^^^^) is larger than the outer diameter of the second beam 124 at the impingementsurface 162 (^^^^^^), which is larger than the outer diameter of the first beam 122 at the impingementsurface 162 (^^^^^^).

[0162] Moreover, the width of the annulus of each of the first, second, and third beams 122, 124,126 at the impingement surface 162 is shown by Equations (12)-(14) :

[0163] In Equations (12)-(14), ^^^^^^^^^^ℎ is the width of the annulus of the first beam 122 at theimpingement surface 162, ^^^^^^^^^^ℎ is the width of the annulus of the second beam 124 at theimpingement surface 162, and ^^^^^^^^^^ℎ is the width of the annulus of the third beam 126 at theimpingement surface 162. Without intending to be limited by theory, and as shown by Equations (12)-(14), in embodiments for forming a defect that is a symmetric C-chamfer shape (i.e.,embodiments in which ^^^^^^^^, ^^^^^^^^, ^^^^^^^^ are equal and the magnitude of chamfer angles ^CH1 and^CH3 are equal), the width of the annulus of the first, second, and third beams 122, 124, 126 are equal as are the cone angles of the first, second, and third beams 122, 124, 126.

[0164] Referring again to FIG. 3, the second lens 132 of the lens assembly 130 may focus eachbeam 122, 124, 126 of the laser beam combination 120 into the transparent workpiece 160, which may be positioned at an imaging plane of this second lens 132. In some embodiments, the first lens 131 and the second lens 132 each comprise plano-convex lenses, meniscus lenses, aspheres, or combinations thereof. In operation, the lens assembly 130 may control the position of the laser beam focal lines 125A, 125B, 125C along the respective beam propagation axes 121A, 121B, 121C of the beams 122, 124, 126. Moreover, each beam 122, 124, 126 may comprise an annular shape when impinging the second lens 132. While the second lens 132 is depicted focusing the laser beam combination 120 into laser beam focal lines 125A, 125B, 125C, other embodimentsAttorney Docket No.: SP24-029_PCT may use the second refractive optical element 144B to both phase modify and focus the laser beam combination 120, for example, without the use of the lens assembly 130.

[0165] While not intending to be limited by theory, after the laser beam combination 120 hasbeen phase modified by the second refractive optical element 144B, each laser beam focal line 125A, 125B, 125C within the transparent workpiece 160 comprises a circular or approximately circular angular spectrum. Each laser beam focal line 125A, 125B, 125C within transparent workpiece 160 has a Rayleigh range defined by a divergence factor FDgreater than or equal to 10. Indeed, the embodiments described herein in which the first beam 122 and the third beam 126 impinge the impingement surface non-orthogonally, the first beam 122 and the third beam 126 are aberrated (that is, have a non-circular angular spectrum) when the beams 122, 126 are upstream the transparent workpiece 160, and upon refraction of the beams 122, 126 at the impingement surface 162 of the transparent workpiece 160, the beams 122, 126 exhibits a quasi-non-diffracting character with minimal to no aberrations within the transparent workpiece 160 (that is, the first and third laser beam focal lines 125A, 125C within transparent workpiece 160 each has a circular or approximately circular angular spectrum). Indeed, the second refractive optical element 144B is configured such that the aberration imposed on each of the first and third beams 122, 126 is the inverse of the aberration that would be imparted to a corresponding unaberrated laser beam incident to impingement surface 162 at the same angle of incidence upon refraction at impingement surface 162 into transparent workpiece 160. As a result, the aberration imposed by the second refractive optical element 144B on the first beam 122 and the third beam 126 is reversed upon refraction of the first beam 122 and the third beam 126 at the impingement surface 162 so that the first beam 122 and the third beam 126 within the transparent workpiece 160 are essentially unaberrated and the first and third laser beam focal lines 125A, 125C have essentially a circular angular spectrum. While the first beam 122 and the third beam 126 are discussed herein as impinging the transparent workpiece 160 non-orthogonally, it should be understood that embodiments are contemplated in which other beams of the laser beam combination 120 impinge the transparent workpiece non-orthogonally. Indeed, the above discussion applies to any beam of the laser beam combination 120 impinging the transparent workpiece 160 non-orthogonally.

[0166] While not intending to be limited by theory, it should be understood that Snell’s lawimposes some limitations on the maximum chamfer angle θCH1, θCH2 of the first and third laserAttorney Docket No.: SP24-029_PCT beam focal lines 125A, 125C formed using the above described techniques (or any laser beam focal lines directed into the transparent workpiece non-orthogonally). Snell’s law ismathematically defined as ^^ = si −12 nwhere ^^1 is the angle of an incident light ray in afirst medium (e.g., air), ^^2 is the angle of the ray in a second medium (e.g., the transparentworkpiece is the index of refraction of the first medium (e.g., air, which comprisesindex of refraction of about 1), and ^^2 is the index of refraction of the second medium (e.g., thetransparent workpiece 160, which may comprise about 1.45 in embodiments in which thetransparent workpiece 160 comprises glass). The angles ^^1 and ^^2 are measured relative to thenormal to the surface of incidence (e.g. impingement surface 162) of the light ray. Snell’s law provides a fundamental limit on the angle of light that can be achieved within the transparent workpiece 160. This limit is the critical angle of the transparent workpiece 160. It should be understood that, for a transparent workpiece 160 comprising another material besides glass, the critical angle would vary based on the index of refraction of that particular material.

[0167] When the transparent workpiece 160 comprises glass having an index of refraction of1.45, the critical angle is about 43.6°. The critical angle is also the internal angle a light ray would take if it contacted the glass with an almost 90° incidence. Thus, Snell’s Law limits the chamfer angles θCH1,θCH2of the first and third laser beam focal lines 125A, 125C. Furthermore, as the beams 122, 124, 126 each comprise a cone shape when impinging the impingement surface 162 of the transparent workpiece 160, and thus comprises a cone angle, which may be from 5° to 30°. As an example, if the first beam 122 impinges the impingement surface 162 of the transparent workpiece 160 with a cone angle of 10°, the maximum chamfer angle θCH1 of the first laser beam focal line 125A inside the transparent workpiece 160 would be 33.6°, assuming light could be incident to the transparent workpiece 160 up to 90°.

[0168] While not intending to be limited by theory, some reflection of the respective beams 122,124, 126 may occur at the impingement surface 162 of the transparent workpiece 160. For example, the reflection of a light ray impinging the impingement surface 162 at 90° relative to normal an impingement location 111A, 111B, 111C will be 100% for both S-polarization and P- polarization and the reflection of a light ray impinging the impingement surface 162 at angles less than 90° relative to normal the impingement location 111A, 111B, 111C will be less than 100%Attorney Docket No.: SP24-029_PCT of S-polarization and P-polarization. While the respective beams 122, 124, 126 may comprise P- polarized light or S-polarized light, P-polarized light may reduce loss due to reflection. For example, at 85 degrees, the reflectance for S-polarized light is 73%, and reflectance for P-polarized light is 49%. In operation, the beam source 10, the first and second refractive optical elements 144A and 144B, or an additional optical component, such as a polarizer, may be used to S-polarize or P-polarize the respective beams 122, 124, 126. While still not intending to be limited by theory, if the magnitude of light intensity around the angular spectrum of the respective laser beam focal lines 125A, 125B, 125C within the transparent workpiece 160 is non-uniform, the respective laser beam focal lines 125A, 125B, 125C retain a circular angular spectrum and a quasi-non-diffracting character within the transparent workpiece 160. However, non-uniform magnitude of light intensity around the angular spectrum of respective laser beam focal lines 125A, 125B, 125C within the transparent workpiece 160 caused by reflection may be compensated for by launching the initial laser beam 12 (i.e., launching the initial laser beam 12 from the beam source 10) with a non-uniform intensity, where the non-uniform intensity is configured to become uniform around the angular spectrum once the light is refracted at the impingement surface 162 and enters the transparent workpiece 160. Example non-uniform intensity beams that may be used (and then converted into a quasi-non-diffracting beam with an oblong or otherwise non-uniform angular spectrum by first and second refractive optical elements 144A and 144B) include an elliptical- Gaussian beam, a top hat beam, or another beam having an arbitrary intensity profile.

[0169] Referring again to FIGS. 1A-3B, in operation, the laser beam combination 120 may betranslated relative to the transparent workpiece 160 (e.g., in the translation direction 101) along the contour line 165 to form the plurality of defects 172 of the contour 170. Directing or localizing the laser beam combination 120 into the transparent workpiece 160 generates an induced absorption (e.g. nonlinear absorption, multiphoton absorption) within the transparent workpiece 160 and deposits enough energy to break chemical bonds in the transparent workpiece 160 at spaced locations along the contour line 165 to form the defects 172. According to one or more embodiments, the laser beam combination 120 may be translated across the transparent workpiece 160 by motion of the transparent workpiece 160 (e.g., motion of a translation stage 109 coupled to the transparent workpiece 160, as shown in FIG. 3), motion of the laser beam combination 120 (e.g., motion of the respective laser beam focal lines 125A, 125B, 125C), or motion of both the transparent workpiece 160 and the respective laser beam focal lines 125A, 125B, 125C.Attorney Docket No.: SP24-029_PCT

[0170] Referring again to FIGS. 1A-13, the defects 172 may generally be spaced apart from oneanother by a distance along the contour 170 of from about 0.1 µm to about 500 µm, for example, about 1 µm to about 200 µm, about 2 µm to about 100 µm, about 5 µm to about 20 µm, or the like. For example, suitable spacing between the defects 172 may be from about 0.1 µm to about 50 µm, such as from about 5 µm to about 15 µm, from about 5 µm to about 12 µm, from about 7 µm to about 15 µm, or from about 7 µm to about 12 µm. In some embodiments, a spacing between adjacent defects 172 may be about 50 µm or less, 45 µm or less, 40 µm or less, 35 µm or less, 30 µm or less, 25 µm or less, 20 µm or less, 15 µm or less, 10 µm or less, or the like.

[0171] As illustrated in FIGS. 1A and 1B, the plurality of defects 172 of the contour 170 extendinto the transparent workpiece 160 and establish a path for crack propagation for separation of the transparent workpiece 160 into separate portions along the contour 170. Forming the contour 170 comprises translating at least one of the laser beam combination 120 and the transparent workpiece 160 relative to one another (e.g., in the translation direction 101) along the contour line 165 to form the plurality of defects 172 of the contour 170. According to one or more embodiments, the laser beam combination 120 may be translated across the transparent workpiece 160 by motion of the transparent workpiece 160, motion of the laser beam combination 120 (e.g., motion of the laser beam focal lines 125A, 125B, 125C), or motion of both the transparent workpiece 160 and the laser beam combination 120, for example, using one or more translation stages 109 (FIGS.2A and 2B). By translating the laser beam focal lines 125A, 125B, 125C relative to the transparent workpiece 160, the plurality of defects 172 may be formed in the transparent workpiece 160, wherein each of the plurality of defects 172 comprising defect segments 172 having a variety of defect angles.

[0172] Suitable laser wavelengths for forming defects 172 are wavelengths at which thecombined losses of linear absorption and scattering by the transparent workpiece 160 are sufficiently low. In embodiments, the combined losses due to linear absorption and scattering by the transparent workpiece 160 at the wavelength are less than 20% / mm, or less than 15% / mm, or less than 10% / mm, or less than 5% / mm, or less than 1% / mm, where the dimension “ / mm” means per millimeter of distance within the transparent workpiece 160 in the beam propagation direction of the laser beam combination 120 (e.g., the Z direction). Representative wavelengths for many glass workpieces include fundamental and harmonic wavelengths of Nd3+(e.g. Nd3+:YAG orAttorney Docket No.: SP24-029_PCT Nd3+:YVO4 having fundamental wavelength near 1064 nm and higher order harmonic wavelengths near 532 nm, 355 nm, and 266 nm). Other wavelengths in the ultraviolet, visible, and infrared portions of the spectrum that satisfy the combined linear absorption and scattering loss requirement for a given substrate material can also be used.

[0173] In operation, the laser beam combination 120 may create multi-photon absorption (MPA)in the transparent workpiece 160. MPA is the simultaneous absorption of two or more photons of identical or different frequencies that excites a molecule from one state (usually the ground state) to a higher energy electronic state (i.e., ionization). The energy difference between the involved lower and upper states of the molecule is equal to the sum of the energies of the involved photons. MPA, also called induced absorption, can be a second-order or third-order process (or higher order), for example, that is several orders of magnitude weaker than linear absorption. It differs from linear absorption in that the strength of second-order induced absorption may be proportional to the square of the light intensity, for example, and thus it is a nonlinear optical process.

[0174] The perforation step that creates the contour 170 (FIGS. 1A and 1B) may utilize the beamsource 10 (e.g., a pulsed beam source such as an ultra-short pulse laser) in combination with the first and second refractive optical elements 144A and 144B, the first lens 131, and the second lens 132, to irradiate the transparent workpiece 160 and generate the laser beam focal lines 125A, 125B, 125C. The laser beam focal lines 125A, 125B, 125C comprise quasi-non-diffracting beams, such as a Gauss-Bessel beam or Bessel beam, as defined above, and may fully or partially perforate the transparent workpiece 160 to form defects 172, with defect segments comprising a variety of defect angles in the transparent workpiece 160, which may form the contour 170. In embodiments in which the laser beam combination 120 comprises a pulsed laser beam, the pulse duration of the individual pulses is in a range of from about 1 femtosecond to about 200 picoseconds, such as from about 1 picosecond to about 100 picoseconds, 5 picoseconds to about 20 picoseconds, or the like, and the repetition rate of the individual pulses may be in a range from about 1 kHz to 4 MHz, such as in a range from about 10 kHz to about 3 MHz, or from about 10 kHz to about 650 kHz. Configuration of Laser Beam

[0175] Referring now to FIGS. 15A and 15B, in addition to a single pulse operation at theaforementioned individual pulse repetition rates, in embodiments comprising a pulsed laser beam,Attorney Docket No.: SP24-029_PCT the pulses may be produced in pulse bursts 500 of two sub-pulses 500A or more (such as, for example, 3 sub-pulses, 4 sub-pulses, 5 sub-pulses, 10 sub-pulses, 15 sub-pulses, 20 sub-pulses, or more per pulse burst, such as from 1 to 30 sub-pulses per pulse burst 500, or from 5 to 20 sub- pulses per pulse burst 500). While not intending to be limited by theory, a pulse burst is a short and fast grouping of sub-pulses that creates an optical energy interaction with the material (i.e. MPA in the material of the transparent workpiece 160) on a time scale not easily accessible using a single-pulse operation. While still not intending to be limited by theory, the energy within a pulse burst (i.e. a group of pulses) is conserved. As an illustrative example, for a pulse burst having an energy of 100 µJ / burst and 2 sub-pulses, the 100 µJ / burst energy is split between the 2 pulses for an average energy of 50 µJ per sub-pulse and for a pulse burst having an energy of 100 µJ / burst and 10 sub-pulses, the 100 µJ / burst is split amongst the 10 sub-pulses for an average energy of 10 µJ per sub-pulse. Further, the energy distribution among the sub-pulses of a pulse burst does not need to be uniform. In fact, in some instances, the energy distribution among the sub-pulses of a pulse burst is in the form of an exponential decay, where the first sub-pulse of the pulse burst contains the most energy, the second sub-pulse of the pulse burst contains slightly less energy, the third sub-pulse of the pulse burst contains even less energy, and so on. However, other energy distributions within an individual pulse burst are also possible, where the exact energy of each sub- pulse can be tailored to effect different amounts of modification to the transparent workpiece 160.

[0176] While still not intending to be limited by theory, when the defects 172 of the one or morecontours 170 are formed with pulse bursts having at least two sub-pulses, the force necessary to separate the transparent workpiece 160 along the contour 170 (i.e. the maximum break resistance) is reduced compared to the maximum break resistance of a contour 170 with the same spacing between adjacent defects 172 in an identical transparent workpiece 160 that is formed using a single pulse laser. For example, the maximum break resistance of a contour 170 formed using a single pulse is at least two times greater than the maximum break resistance of a contour 170 formed using a pulse burst having 2 or more sub-pulses. Further, the difference in maximum break resistance between a contour 170 formed using a single pulse and a contour 170 formed using a pulse burst having 2 sub-pulses is greater than the difference in maximum break resistance between a contour 170 formed using a pulse burst having 2 sub-pulses and a pulse burst having 3 sub- pulses. Thus, pulse bursts may be used to form contours 170 that separate easier than contours 170 formed using a single pulse laser.Attorney Docket No.: SP24-029_PCT

[0177] Referring still to FIGS. 15A and 15B, the sub-pulses 500A within the pulse burst 500may be separated by a duration that is in a range from about 1 nsec to about 50 nsec, for example, from about 10 nsec to about 30 nsec, such as about 20 nsec. In other embodiments, the sub-pulses 500A within the pulse burst 500 may be separated by a duration of up to 100 psec (for example, 0.1 psec, 5 psec, 10 psec, 15 psec, 18 psec, 20 psec, 22 psec, 25 psec, 30 psec, 50 psec, 75 psec, or any range therebetween). For a given laser, the time separation Tpbetween adjacent sub-pulses 500A within a pulse burst 500 may be relatively uniform (e.g., within about 10% of one another). For example, in some embodiments, each sub-pulse 500A within a pulse burst 500 is separated in time from the subsequent sub-pulse by about 20 nsec (50 MHz). Further, the time between each pulse burst 500 may be from about 0.25 microseconds to about 1000 microseconds, e.g., from about 1 microsecond to about 10 microseconds, or from about 3 microseconds to about 8 microseconds.

[0178] In some of the exemplary embodiments of the beam source 10 described herein, the timeseparation Tbis about 5 microseconds for the beam source 10 outputting an initial laser beam 12 comprising a burst repetition rate of about 200 kHz. The laser burst repetition rate is related to the time Tb between the first pulse in a burst to the first pulse in the subsequent burst (laser burst repetition rate = 1 / Tb). In some embodiments, the laser burst repetition rate may be in a range of from about 1 kHz to about 4 MHz. In embodiments, the laser burst repetition rates may be, for example, in a range of from about 10 kHz to 650 kHz. The time Tb between the first pulse in each burst to the first pulse in the subsequent burst may be from about 0.25 microsecond (4 MHz burst repetition rate) to about 1000 microseconds (1 kHz burst repetition rate), for example from about 0.5 microseconds (2 MHz burst repetition rate) to about 40 microseconds (25 kHz burst repetition rate), or from about 2 microseconds (500 kHz burst repetition rate) to about 20 microseconds (50k Hz burst repetition rate). The exact timing, pulse duration, and burst repetition rate may vary depending on the laser design, but short pulses (Td <20 psec and, in some embodiments, Td 15 psec) of high intensity have been shown to work particularly well.

[0179] The burst repetition rate may be in a range of from about 1 kHz to about 2 MHz, such asfrom about 1 kHz to about 200 kHz. Bursting or producing pulse bursts 500 is a type of laser operation where the emission of sub-pulses 500A is not in a uniform and steady stream but rather in tight clusters of pulse bursts 500. The pulse burst laser beam may have a wavelength selectedAttorney Docket No.: SP24-029_PCT based on the material of the transparent workpiece 160 being operated on such that the material of the transparent workpiece 160 is substantially transparent at the wavelength. The average laser power per burst measured at the material may be at least about 40 µJ per mm of thickness of material. For example, in embodiments, the average laser power per burst may be from about 40 μJ / mm to about 2500 μJ / mm, or from about 500 μJ / mm to about 2250 μJ / mm. In a specific example, for 0.5 mm to 0.7 mm thick Corning EAGLE XG®transparent workpiece, pulse bursts of from about 300 μJ to about 600 µJ may cut and / or separate the workpiece, which corresponds to an exemplary range of about 428 µJ / mm to about 1200 µJ / mm (e.g., 300 µJ / 0.7 mm for 0.7 mm EAGLE XG®glass and 600 µJ / 0.5 mm for a 0.5 mm EAGLE XG®glass).

[0180] The energy required to modify the transparent workpiece 160 is the pulse energy, whichmay be described in terms of pules burst energy (i.e., the energy contained within a pulse burst 500 where each pulse burst 500 contains a series of sub-pulses 500A), or in terms of the energy contained within a single laser pulse (many of which may comprise a burst). The pulse energy (for example, pulse burst energy) may be from about 25 µJ to about 750 µJ, e.g., from about 50 µJ to about 500 µJ, or from about 50 µJ to about 250 µJ. For some glass compositions, the pulse energy (e.g., pulse burst energy) may be from about 100 µJ to about 250 µJ. However, for some glass compositions, the pulse energy (e.g., pulse burst energy) may be higher (e.g., from about 300 µJ to about 500 µJ, or from about 400 µJ to about 600 µJ, depending on the specific glass composition of the transparent workpiece 160).

[0181] The portion of the first, second, and third beams 122, 124, 126 directed into thetransparent workpiece 160 may comprise a persistent intensity homogeneity. As used herein, the term “persistent intensity homogeneity” means that an intensity of the laser beam at any point within any of the laser beam focal lines 125A, 125B, and 125C does not vary by more than 50% from the average intensity assessed collectively over the laser beam focal lines 125A, 125B, and 125C. In the embodiments described in this disclosure, the persistent intensity homogeneity of the laser beam throughout the portion of the laser beam focal region within the transparent workpiece is such that the intensity for any point within any of the focal lines present in the transparent workpiece, the extrema (i.e., the minimum or maximum) of the intensity of the laser beam is greater than or equal to about is between 50% and 150% of the average intensity assessed collectively over all of the focal lines in the transparent workpiece.Attorney Docket No.: SP24-029_PCT

[0182] While not intending to be limited by theory, the use of a pulsed laser beam capable ofgenerating pulse bursts is advantageous for cutting or modifying transparent materials, for example glass (e.g., the transparent workpiece 160). In contrast with the use of single pulses spaced apart in time by the repetition rate of the single-pulsed laser, the use of a burst sequence that spreads the pulse energy over a rapid sequence of pulses within the burst allows access to larger timescales of high intensity interaction with the material than is possible with single-pulse lasers. The use of pulse bursts (as opposed to a single pulse operation) increases the size (e.g., the cross-sectional size) of the defects 172, which facilitates the connection of adjacent defects 172 when separating transparent workpiece 160 along the one or more contours 170, thereby minimizing unintended crack formation. Further, using a pulse burst to form defects 172 increases the randomness of the orientation of cracks extending outward from each defect 172 into the bulk material of the transparent workpiece 160 such that individual cracks extending outward from defects 172 do not influence or otherwise bias the separation of the contour 170 such that separation of the defects 172 follows the contour 170, minimizing the formation of unintended cracks.

[0183] An example intensity distribution of a laser beam focal line over distance is shown inFIG.15C. FIG.15C demonstrates generation of a Bessel beam using a Gaussian laser input. When a Gaussian input beam is passed through the first and second refractive optical elements described herein, the characteristic maximum intensity distribution of the Bessel beam with respect to Z initially rises due to an increase in Gaussian beam radius and then gradually falls off due to reduced intensity of the Gaussian input at larger radii. One way to counteract this asymmetry is to place the impingement surface 162 of the transparent workpiece 160 greater than or equal to 0.05 mm, greater than or equal to 0.08 mm, greater than or equal to 0.10 mm, greater than or equal to 0.12 mm, greater than or equal to 0.15 mm, greater than or equal to 0.18 mm, or greater than or equal to 0.20 mm downstream from the formation of the first laser beam focal line 125A, such that the first laser beam focal line 125A comprises a portion external the transparent workpiece 160. Placing the impingement surface 162 of the transparent workpiece 160 downstream from the formation of the first laser beam focal line 125A may result in a greater beam intensity at the impingement surface 162, ensuring that the internal defect plane of the first defect segment 172A connects with the impingement surface 162. Another way to counteract the asymmetry shown in FIG.15C is to modify the initial laser beam 12 from the standard Gaussian shape, as described in international patent application publication no. WO 2023 / 009331 A1.Attorney Docket No.: SP24-029_PCT

[0184] Referring again to FIG. 3, the optical assembly 100 may be configured to further alterthe laser beam combination 120 such that a cross-section of each beam 122, 124, 126 at the impingement surface 162 of the transparent workpiece 160 is non-axisymmetric and thus a cross- section of each laser beam focal line 125A, 125B, 125C is non-axisymmetric, for example, using the methods and systems described in U.S. Pat. No.10,730,783, hereby incorporated by reference in its entirety. For example, the beam spots 112A, 112B, 112C formed by the beams 122, 124, 126 at the impingement surface 162 the transparent workpiece 160 may comprise a non- axisymmetric beam spot having a long axis and a short axis such that the defect segments 172A, 172B, 172C formed using laser beam focal lines 125A, 125B, 125C comprise a central defect region formed at the intersection of the long axis and the short axis and one or more radial arms formed in the direction of the long axis. Defect segments 172A, 172B, 172C formed using laser beam focal lines 125A, 125B, 125C having a non-axisymmetric beam spot oriented such that the long axis of each beam spot 112A, 112B, 112C extends along the contour line 165 thereby forming defects 172 with radial arms that extend along the contour line 165. By controlling the laser beam focal lines 125A, 125B, 125C such that the direction of the radial arms of each defect 172 extends along the contour line 165, crack propagation may be better controlled. The laser beam focal lines 125A, 125B, 125C with cross-sections that are non-axisymmetric may be formed by altering the beamlet regions of the second refractive optical element 144B, blocking a portion of each beam 122, 124, 126 of the laser beam combination 120.

[0185] Referring again to FIGS. 1A-15C, in some embodiments, the transparent workpiece 160may be further acted upon in a subsequent separating step to induce separation of the transparent workpiece 160 along the contour 170 to form a separated transparent article comprising an angled edge (FIGS. 16A-16B). The subsequent separating step may include using mechanical force, thermal stress induced force, or a chemical etchant to propagate a crack along the contour 170. The thermal source, such as an infrared laser beam, may be used to create thermal stress and thereby separate the transparent workpiece 160 along the contour 170. Separating the transparent workpiece 160 may include directing an infrared laser beam at the contour 170 to induce thermal stress to propagate a crack along the contour 170. In some embodiments, the infrared laser beam may be used to initiate separation and then the separation may be finished mechanically. Without being bound by theory, the infrared laser is a controlled heat source that rapidly increases the temperature of the transparent workpiece 160 at or near the contour 170. This rapid heating mayAttorney Docket No.: SP24-029_PCT build compressive stress in the transparent workpiece 160 on or adjacent to the contour 170. Since the area of the heated glass surface is relatively small compared to the overall surface area of the transparent workpiece 160, the heated area cools relatively rapidly. The resultant temperature gradient induces tensile stress in the transparent workpiece 160 sufficient to propagate a crack along the contour 170 and through the depth of the transparent workpiece 160, resulting in full separation of the transparent workpiece 160 along the contour 170. Without being bound by theory, it is believed that the tensile stress may be caused by expansion of the glass (i.e., changed density) in portions of the workpiece with higher local temperature.

[0186] Suitable infrared lasers to create thermal stress in glass would typically have wavelengthsthat are readily absorbed by glass, typically having wavelengths ranging from 1.2 µm to 13 µm, for example, a range of 4 µm to 12 µm. Further, the power of the infrared laser beam may be from about 10 W to about 1000 W, for example 100 W, 250 W, 500 W, 750 W, or the like. Moreover, the 1 / e2beam diameter of the infrared laser beam may be about 20 mm or less, for example, 15 mm, 12 mm, 10 mm, 8 mm, 5 mm, 2 mm, or less. In operation, a larger 1 / e2beam diameter of the infrared laser beam may facilitate faster laser processing and more power while a smaller 1 / e2beam diameter of the infrared laser beam may facilitate high precision separation by limiting damage to portions of the transparent workpiece 160 near the contour 170. Example infrared lasers include a carbon dioxide laser (a “CO2 laser”), a carbon monoxide laser (a “CO laser”), a solid state laser, a laser diode, or combinations thereof.

[0187] In other embodiments, stress present in the transparent workpiece 160, depending on thetype, depth, and material properties (e.g., absorption, CTE, stress, composition, etc.) may cause spontaneous separation along the contour 170 without further heating or mechanical separation steps. For example, when the transparent workpiece 160 comprises a strengthened glass substrate (e.g., an ion-exchanged or thermally tempered glass substrate), the formation of the contour 170 may induce crack propagation along the contour 170 to separate the transparent workpiece 160.

[0188] Referring now to FIGS. 16A and 16B an example transparent workpiece 260 and aresultant separated article 260' formed from the transparent workpiece 260 using the methods and systems described herein are schematically depicted. As one example, FIG. 16A depicts a schematic side view of a transparent workpiece 260 comprising a plurality of defects 272,Attorney Docket No.: SP24-029_PCT including a first defect 272a, a second defect 272b, and a third defect 272c. The first defect 272a extends from an impingement surface 262 to a first end of the second defect 272b, the second defect 272b extends from an end of the first defect 272a to an end of the third defect 272c, and the third defect 272c extends from a second end of the second defect 272b to the second surface 264. In operation, the transparent workpiece 260 may be separated along the plurality of defects 272 using the embodiments described herein to form a separated article 260' having a C-chamfered edge 268, as depicted in FIG.12B.

[0189] For the purposes of describing and defining the present inventive technology, it is notedthat reference herein to a variable being a “function” of a parameter or another variable is not intended to denote that the variable is exclusively a function of the listed parameter or variable. Rather, reference herein to a variable that is a “function” of a listed parameter is intended to be open ended such that the variable may be a function of a single parameter or a plurality of parameters.

[0190] As used herein, the term “about” means that amounts, sizes, formulations, parameters,and other quantities and characteristics are not and need not be exact, but may be approximate and / or larger or smaller, as desired, reflecting tolerances, conversion factors, rounding off, measurement error and the like, and other factors known to those of skill in the art. When the term “about” is used in describing a value or an end-point of a range, the specific value or end-point referred to is included. Whether or not a numerical value or end-point of a range in the specification recites “about,” two embodiments are described: one modified by “about,” and one not modified by “about.” It will be further understood that the endpoints of each of the ranges are significant both in relation to the other endpoint, and independently of the other endpoint.

[0191] Directional terms as used herein- for example up, down, right, left, front, back, top,bottom - are made only with reference to the figures as drawn and are not intended to imply absolute orientation.

[0192] Unless otherwise expressly stated, it is in no way intended that any method set forthherein be construed as requiring that its steps be performed in a specific order, nor that with any apparatus specific orientations be required. Accordingly, where a method claim does not actually recite an order to be followed by its steps, or that any apparatus claim does not actually recite anAttorney Docket No.: SP24-029_PCT order or orientation to individual components, or it is not otherwise specifically stated in the claims or description that the steps are to be limited to a specific order, or that a specific order or orientation to components of an apparatus is not recited, it is in no way intended that an order or orientation be inferred, in any respect. This holds for any possible non-express basis for interpretation, including: matters of logic with respect to arrangement of steps, operational flow, order of components, or orientation of components; plain meaning derived from grammatical organization or punctuation, and; the number or type of embodiments described in the specification.

[0193] As used herein, the singular forms “a,” “an” and “the” include plural referents unless thecontext clearly dictates otherwise. Thus, for example, reference to “a” component includes aspects having two or more such components, unless the context clearly indicates otherwise.

[0194] It will be apparent to those skilled in the art that various modifications and variations canbe made to the embodiments described herein without departing from the spirit and scope of the claimed subject matter. Thus it is intended that the specification cover the modifications and variations of the various embodiments described herein provided such modification and variations come within the scope of the appended claims and their equivalents.

Claims

Attorney Docket No.: SP24-029_PCT CLAIMS What is claimed is:

1. An optical system comprising:a first refractive optical element configured to separate a laser beam into a plurality of beamlets each propagating in a different propagation direction, a second refractive optical element comprising a plurality of beamlet regions, each beamlet region being configured to apply a different phase distribution to a separate one of the plurality of beamlets to convert the plurality of beamlets into a plurality of quasi-non-diffracting laser beams, wherein: each of the plurality of quasi-non-diffracting laser beams propagates along a separate propagation axis, and at least 50% of areas of phase distributions applied individually by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2.

2. The optical system of claim 1, wherein the first refractive optical element comprises aplurality of concentric regions configured to separate different spatial portions of the laser beam into the plurality of beamlets.

3. The optical system of claim 2, wherein the plurality of concentric regions comprises:a central region configured to generate a first beamlet from a central portion of the laser beam, a first annular region configured to generate a second beamlet from a first annular portion of the laser beam circumferentially surrounding the central portion, and a second annular region configured to generate a third beamlet from a second annular portion of the laser beam circumferentially surrounding the first annular portion.

4. The optical system of any of claims 2-3, wherein each of the plurality of concentricregions applies a different linear phase to each of the different spatial portions so that each of theAttorney Docket No.: SP24-029_PCT plurality of beamlets is incident on one of the beamlet regions of the second refractive optical element.

5. The optical system of claim 4, wherein centers of each of the plurality of beamlet regionsform a line in the transverse optical plane.

6. The optical system of claim 5, wherein the plurality of beamlet regions comprises two ormore beamlet regions having non-equal surface areas.

7. The optical system of claim 6, wherein the two or more beamlet regions comprises:a first beamlet region having a smallest projected surface area of the plurality of beamlet regions and disposed at a first end of the plurality of beamlet regions, a third beamlet region having a largest projected surface area of the plurality of beamlet regions and disposed at a second end of the plurality of beamlet regions, and a second beamlet region being disposed between the first beamlet region and the third beamlet region and having a projected surface area between that of the first beamlet region and the third beamlet region, wherein the plurality of concentric regions are configured such that the central portion is directed to the first beamlet region, the first annular portion is directed to the second beamlet region, and the second annular portion is directed to the third beamlet region.

8. The optical system of claim 7, wherein:the second beamlet region is oriented within 1° of normal relative to the optical axis, and the first beamlet region and the third beamlet region are inclined with respect to the second beamlet region at inclination angles of at least 0.25°.

9. The optical system of any of claims 1-8, wherein the first refractive optical element isconstructed such that each of the plurality of beamlets comprises substantially equal power.

10. The optical system of any one of claims 1-9, wherein each of the plurality of beamletregions is configured to apply a corrective phase configured to counteract the phase imparted by the first refractive optical element and a quasi-non-diffractive phase to each beamlet.Attorney Docket No.: SP24-029_PCT 11. The optical system of any one of claims 1-10, wherein at least one of the plurality ofbeamlet regions are configured to apply a linear phase to at least one of the plurality of beamlets so that at least two of the plurality of quasi-non-diffracting beams propagate in different propagation directions downstream of the second refractive optical element.

12. The optical system of any one of claims 1-11, further comprising a lens assemblyconfigured to relay the plurality of quasi-non-diffracting beams downstream from the first refractive optical element and the second refractive optical element.

13. The optical system of claim 12, further comprising a transparent workpiece disposeddownstream from the lens assembly, wherein the plurality of focal lines are formed in the transparent workpiece and generate defects extending through the thickness of the transparent workpiece at angles relative to one another.

14. The optical system of any of claims 1-13, wherein the first refractive optical element andthe second refractive optical element are disposed in a common housing configured to rotate about an optical axis of the optical system to rotate the orientation of the plurality of quasi-non- diffracting laser beams.

15. An optical system for forming defects in a transparent workpiece, the optical systemcomprising: a laser beam source configured to emit a laser beam; a first refractive optical element configured to separate the laser beam into a plurality of beamlets, a second refractive optical element comprising a plurality of beamlet regions, each beamlet region being positioned to receive one of the plurality of beamlets and apply a phase distribution thereto to convert that beamlet into a quasi-non-diffracting laser beam, wherein at least 50% of areas of phase distributions applied individually by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2; andAttorney Docket No.: SP24-029_PCT a lens system configured to relay each quasi-non-diffracting laser beam into the transparent workpiece to generate defects with defect segments that extend through the transparent workpiece at angles relative to one another in a defect plane.

16. The optical system of claim 15, wherein the first refractive optical element comprises aplurality of concentric regions configured to separate different spatial portions of the laser beam into the plurality of beamlets.

17. The optical system of claim 16, wherein the plurality of concentric regions comprises:a central region configured to generate a first beamlet from a central portion of the laser beam, a first annular region configured to generate a second beamlet from a first annular portion of the laser beam circumferentially surrounding the central portion, and a second annular region configured to generate a third beamlet from a second annular portion of the laser beam circumferentially surrounding the first annular portion.

18. The optical system of any of claims 16-17, wherein each of the plurality of concentricregions applies a different linear phase to each of the different spatial portions so that each of the plurality of beamlets is incident on one of the beamlet regions of the second refractive optical elements.

19. The optical system of claim 18, wherein centers of each of the plurality of beamletregions form a line in the transverse optical plane.

20. The optical system of claim 19, wherein the plurality of beamlet regions comprises:a first beamlet region having a smallest projected surface area of the plurality of beamlet regions and disposed at a first end of the plurality of beamlet regions, a third beamlet region having a largest projected surface area of the plurality of beamlet regions and disposed at a second end of the plurality of beamlet regions, andAttorney Docket No.: SP24-029_PCT a second beamlet region being disposed between the first beamlet region and the third beamlet region and having a projected surface area between that of the first beamlet region and the third beamlet region.

21. The optical system of claim 20, wherein the plurality of concentric regions are configuredsuch that the central portion is directed to the first beamlet region, the first annular portion is directed to the second beamlet region, and the second annular portion is directed to the third beamlet region.

22. The optical system of claim 21, wherein:the second beamlet region is oriented within 1° of normal relative to the optical axis, and the first beamlet region and the third beamlet region are inclined with respect to the third beamlet region at inclination angles of at least 0.25°.

23. The optical system of any of claims 15-22, wherein the first refractive optical element isconstructed such that each of the plurality of beamlets comprises substantially equal power.

24. The optical system of any one of claims 15-23, wherein each of the plurality of beamletregions is configured to apply a corrective phase configured to counteract the phase imparted by the first refractive optical element and a quasi-non-diffractive phase to each beamlet.

25. The optical system of any one of claims 15-24, wherein at least one of the plurality ofbeamlet regions are configured to apply a linear phase to at least one of the plurality of beamlets so that at least two of the plurality of quasi-non-diffracting beams propagate in different propagation directions downstream of the second refractive optical element.

26. The optical system of any one of claims 15-25, wherein:the first refractive optical element, the second optical element, and the lens assembly are configured to generate a first quasi-non-diffracting laser beam, a second quasi-non-diffracting laser beam, and a third quasi-non-diffracting laser beam in the transparent workpiece,Attorney Docket No.: SP24-029_PCT the first quasi-non-diffracting beam generates a first defect segment in the defect plane that extends from an impingement surface into the transparent workpiece at a first chamfer angle relative to a surface normal of the impingement surface, the second quasi-non-diffracting laser beam generates a second defect segment in the defect plane that extends from a termination point of the first defect into the transparent workpiece at a second chamfer angle relative to the surface normal, and the third quasi-non-diffracting laser beam generates a third defect segment in the defect plane that extends from a termination point of the second defect at a third chamfer angle relative to the surface normal.

27. The optical system of claim 26, wherein the first chamfer angle equals the third chamferangle and the second chamfer angle is less than 1°.

28. The optical system of claim 27, wherein the first chamfer angle and the third chamferangle are greater than 5°.

29. The optical system of any of claims 15-28, wherein the first refractive optical elementand the second refractive optical element are disposed in a common housing configured to rotate about an optical axis of the optical system to rotate the defect plane within the transparent workpiece.

30. A method for processing a transparent workpiece, the method comprising:generating a plurality of beamlets from a laser beam by transmitting the laser beam through a first refractive optical element; generating a plurality of quasi-non-diffracting laser beams from the plurality of beamlets by transmitting the plurality of beamlets through a second refractive optical element, wherein at least 50% of areas of phase distributions individually applied by each of the first refractive optical element and the second refractive optical element to light from the laser beam are associated with a change in phase slope that is less than 1.7x10-4rad / µm2; directing the plurality of quasi-non-diffracting laser beams into the transparent workpiece simultaneously at different impingement locations on an impingement surface to form a pluralityAttorney Docket No.: SP24-029_PCT of defects, with each defect comprising a plurality of defect segments extending at angles relative to each other in a defect plane.

31. The method of claim 30, further comprising translating the plurality of quasi-diffractinglaser beams and the transparent workpiece relative to each other to move the defect plane and form a plurality of defects along a contour.

32. The method of claim 31, further comprising rotating the first refractive optical elementand the second refractive optical element in conjunction with one another to rotate a linear arrangement of the plurality of quasi-non-diffracting laser beams and provide the contour with a curved shape.

33. The method of any of claims 30-32, further comprising separating the transparentworkpiece along the contour to form a separated glass article.

34. The method of any of claims 30-33, wherein the plurality of quasi-non-diffracting laserbeams forms a first laser beam focal line, a second laser beam focal line, and a third laser beam focal line in the transparent workpiece to generate the plurality of defects, wherein: the first laser beam focal line generates a first defect in the defect plane that extends from an impingement surface into the transparent workpiece at a first chamfer angle relative to a surface normal of the impingement surface, the second laser beam focal line generates a second defect in the defect plane that extends from a termination point of the first defect into the transparent workpiece at a second chamfer angle relative to the surface normal, and the third laser beam focal line generates a third defect in the defect plane that extends from a termination point of the second defect at a third chamfer angle relative to the surface normal.

35. The method of claim 34, wherein the first chamfer angle equals the third chamfer angleand the second chamfer angle is less than 1°.Attorney Docket No.: SP24-029_PCT 36. The method of claim 35, wherein the first chamfer angle and the third chamfer angle aregreater than 5°.

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